Vieta did not approve of the term *tangent* because it could be
confused with the term in geometry. He used *Prosinus* instead.

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 an 1864 *Webster* dictionary (OED2).

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 **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*).

**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)The earliest use of^{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.

Q = SQ + VQ = TQ [times] UQIn his paper "Researches respecting quaternions" (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, themodulus,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 thescalar of a sum is the sum of the scalarsand thevector of the sum is the sum of the vectors,so thattensor of a product is the product of the tensorsand theversor of a product is the product of the versors.In other words, the

tensorof a quaternion is simply its modulus.

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 Einstein in 1916 (Kline,
page 1123).

The term **TERAGON** was coined by Mandelbrot, according to an Internet
web page.

**TERMINATING DECIMAL** appears in *Webster's New International
Dictionary* (1909).

**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).

**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).

The phrase **THEORY OF GAMES** appears in 1943 in the title *Theory
of Games and Economic Behavior* by John von Neumann and Oskar Morgenstern
[James A. Landau].

The term **THRACKLE** was coined by John Horton Conway.

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 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." The
word 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.

**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.

The term **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.

The **TRACTRIX** was named by Christiaan Huygens (1629-1695), according
to the University of St. Andrews website.

**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."

**TRANSCENDENTAL NUMBER.** 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).

The earliest citation in the OED2 for *transcendental* describing
a number is in 1843 in the *Penny Cyclopedia,* which refers to the
roots of an equation as transcendental.

**TRANSFINITE.** Georg Cantor (1845-1918) used this word in the title
of a paper published in 1895, *Beiträge zur Begründung der Transfiniten
Mengenlehre.*

**TRANSPOSE** (of a matrix) is found 1937 in *Mod. Higher Algebra*
by A. A. Albert (OED2).

**TRANSVERSAL** is dated ca. 1847 in MWCD10.

**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:

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).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 atrapezoid.

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.

**TRIANGLE INEQUALITY** appears in 1941 in *Survey of Modern Algebra*
by Birkhoff and MacLane (OED2).

**TRIANGULAR** (referring to a number) appears in English in 1706
in *Synopsis Palmariorum Matheseos* by William Jones (OED2).

The **TRIDENT** was named by Isaac Newton, according to the University
of St. Andrews website. The name Descartes appears in parentheses at the
end of the *trident* definition in *Webster's New International
Dictionary* (1909).

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 (1707-775) "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).

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 letters written in 1670 by Sir
Isaac Newton [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 2possible truth-configurations of the^{n}p's a definite truth-value offis determined. The relation thus effected we shall call the truth-table off.

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 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].

The term **TYPE I ERROR** was used by Jerzy Neyman and Egon S. Pearson
(Kramer, p. 322).