早期數學字彙的歷史 (T)

Last revision: July 14, 1999

TANGENT (in trigonometry) 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 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)2 = (SQ)2 - (VQ)2, and if we suppose TQ to be always a real and positive or absolute number, which we may call the tensor of the quaternion Q, we shall not thereby diminish the generality of that quaternion. This tensor is what was called in former articles the modulus.
The earliest use of tensor in the Proceedings of the Royal Irish Academy is on p. 282 of Volume 3, and is in the proceedings of the meeting held on July 20, 1846. The volume appeared in 1847. Hamilton writes:
Q = SQ + VQ = TQ [times] UQ

The factor TQ is always a positive, or rather an absolute (or signless) number; it is what was called by the author, in his first communication on this subject to the Academy, the modulus, but which he has since come to prefer to call it the TENSOR of the quaternion Q: and he calls the other factor UQ the VERSOR of the same quaternion. As the scalar of a sum is the sum of the scalars and the vector of the sum is the sum of the vectors, so that tensor of a product is the product of the tensors and the versor of a product is the product of the versors.

In other words, the tensor of a quaternion is simply its modulus.

In his paper "Researches respecting quaternions" (Transactions of the Royal Irish Academy, vol. 21 (1848) pp. 199-296), Hamilton uses the term "modulus," not "tensor." This paper purports to have been read on 13 November 1843 (i.e., at the same meeting as the short paper, or abstract, in the Proceedings of the RIA).

The terms vector, scalar, tensor and versor appear in the series of papers "On Quaternions" that appeared in the Philosophical Magazine (see pages 236-7 in vol III of "The Mathematical Papers of Sir William Rowan Hamilton," edited by H. Halberstam and R.E. Ingram). The editors have taken 18 short papers published in the Philosophical Magazine between 1844 and 1850, and concatenated them in the "Mathematical Papers" to form a seamless whole, with no indication as to how the material was distributed into the individual papers.

(Information for this article was provided by David Wilkins and Julio Gonz嫮ez Cabill鏮.)

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 Gonz嫮ez Cabill鏮).

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 "Studi螚 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 Grundzge der Mengenlehre (1914).

TOPOLOGY was introduced in 1847 by Johann Benedict Listing (1808-1882) in "Vorstudien zur Topologie," Vandenhoeck und Ruprecht, G飆tingen, 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, Beitr輍e zur Begrndung 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:

Trapezium...a plane figure contained under four right lines, of which both the opposite pairs are not parallel. When this figure has two of its sides parallel to each other, it is sometimes called a trapezoid.
No previous use the words with Hutton's definitions is known. Nevertheless, the newer meanings of the two words now prevail in U. S. but not necessarily in Great Britain (OED2).

Some geometry textbooks define a trapezoid as a quadrilateral with at least one pair of parallel sides, so that a parallelogram is a type of trapezoid.

TRAVELING SALESMAN PROBLEM. The first use of this term "may have been in 1931 or 1932, when A. W. Tucker heard the term from Hassler Whitney of Princeton University." This information comes from an Internet web page, which refers to E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, D. B. Shmoys, editors, The Traveling Salesman Problem (1985).

Other terms are knight's tour and the messenger problem.

The term TREE in graph theory was coined by James Joseph Sylvester, according to an Internet web site.

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 Klgel (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 methodic 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 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.
TSCHIRNHAUS' CUBIC appears in R. C. Archibald's paper written in 1600 where he attempted to classify curves, according to the University of St. Andrews website.

The term TURING MACHINE was used for the first time in 1937 by Stephen C. Kleene in the Journal of Symbolic Logic, according to an Internet website, which also states that the term Turing test seems to have appeared in the 1970s.

The term TWIN PRIME was coined in 1916 by Paul Gustav St踄kel (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).

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