早期數學字彙的歷史 (I)

Last revision: Aug. 2, 1999


ICOSAHEDRON is found in English in Sir Henry Billingsley's 1570 translation of Euclid's Elements (OED2).

The term IDEAL (as a noun) was introduced by Richard Dedekind (1831-1916) in P. G. L. Dirichlet Vorles. ber Zahlentheorie (ed. 2, 1871) Suppl. x. 452 (OED2).

IDEAL NUMBER. Ernst Eduard Kummer (1810-1893) introduced the term ideale zahl in 1846 in Ber. ber die zur Bekanntmachung geeigneten Verh. d. K. Preuss. Akad. d. Wiss. zu Berlin 87 (OED2).

IDEMPOTENT and NILPOTENT were used by Benjamin Peirce (1809-1880) in 1870:

When an expression raised to the square or any higher power vanishes, it may be called nilpotent; but when, raised to a square or higher power, it gives itself as the result, it may be called idempotent.
The defining equation of nilpotent and idempotent expressions are respectively An = 0, and An = A; but with reference to idempotent expressions, it will always be assumed that they are of the form
A2 = A,
unless it be otherwise distinctly stated.
This citation is excerpted from "Linear Associative Algebra," a memoir read by Benjamin Peirce before the National Academy of Sciences in Washington, 1870, and published by him as a lithograph in 1870. In 1881, Peirce's son, Charles S. Peirce, reprinted it in the American Journal of Mathematics. [Julio Gonz嫮ez Cabill鏮]

The OED2 shows a 1937 citation with a simplified definition of idempotent in Modern Higher Algebra (1938) iii 88 by A. A. Albert: "A matrix E is called idempotent if E2 = E. [Older dictionaries pronounce idempotent with the only stress on the second syllable, but newer ones show a primary stress on the first syllable and a secondary stress on the penult.]

IDENTITY (an equation satisfied by all values of the variable) is found in 1859 in Arithmetic & Algebra in their principles and application, sixth edition, by Barnard Smith: "Such an expression as (x + 1)2 = x2 + 2x + 1, where one of the quantities, between which the sign equality is placed, results from performing the operations indicated in the other, is called an Identity" (OED2).

IDENTITY (the element which leaves other elements unchanged) is found in 1894 in the Bulletin of the American Mathematical Society (OED2).

IDENTITY ELEMENT is found in 1902 in Transactions of the American Mathematical Society III. 486: "There exists a left-hand identity element, that is, an element ile such that, for every element a, ila = a" (OED2).

IDENTITY MATRIX is found in "Representations of the General Symmetric Group as Linear Groups in Finite and Infinite Fields," Leonard Eugene Dickson, Transactions of the American Mathematical Society, Vol. 9, No. 2. (Apr., 1908).

The term is also found in "Concerning Linear Substitutions of Finite Period with Rational Coefficients," Arthur Ranum, Transactions of the American Mathematical Society, Vol. 9, No. 2. (Apr., 1908).

IFF. On the last page of his autobiography, Paul R. Halmos (1916- ) writes:

My most nearly immortal contributions are an abbreviation and a typographical symbol. I invented "iff", for "if and only if" -- but I could never believe that I was really its first inventor. I am quite prepared to beieve that it existed before me, but I don't know that it did, and my invention (re-invention?) of it is what spread it thorugh the mathematical world. The symbol is definitely not my invention -- it appeared in popular magazines (not mathematical ones) before I adopted it, but, once again, I seem to have introduced it into mathematics. It is the symbol that sometimes looks like [an empty square], and is used to indicate an end, usually the end of a proof. It is most frequently called the "tombstone", but at least one generous author referred to it as the "halmos".
This quote is from I Want to Be a Mathematician: An Automathography, by Paul R. Halmos, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985, page 403.

The earliest citation of "iff" in the OED2 is 1955 in General Topology by J. L. Kelley, who credits the term to Halmos.

The terms IMAGINARY and REAL were introduced in French by Rene Descartes (1596-1650) in "La Geometrie" (1637):

...tant les vrayes racines que les fausses ne sont pas tousiours r嶪lles; mais quelquefois seulement imaginaires; c'est dire qu'on peut bien toujiours en imaginer autant que aiy dit en ch跴que Equation; mais qu'il n'y a quelquefois aucune quantit, qui corresponde celles qu'on imagine. comme encore qu'on en puisse imaginer trois en celle cy, x3 - 6xx + 13x - 10 = 0, il n'y en a toutefois qu'une r嶪lle, qui est 2, & pour les deux autres, quois qu'on les augmente, ou diminu, ou multipli en la fa蔞n que ie viens d'憖pliquer, on ne s蓷uroit les rendre autres qu'imaginaires. [...neither the true roots nor the false are always real; sometimes they are, however, imaginary; namely, whereas we can always imagine as many roots for each equation as I have predicted, there is still not always a quantity which corresponds to each root so imagined. Thus, while we may think of the equation x3 - 6xx + 13x - 10 = 0 as having three roots, yet there is just one real root, which is 2, and the other two, however, increased, diminished, or multiplied them as we just laid down, remain always imaginary.] (page 380)
An early appearance of the word imaginary in English is in "A treatise of algebra, both historical and practical" (1685) by John Wallis (1616-1703):
We have before had occasion (in the Solution of some Quadratick and Cubick Equations) to make mention of Negative Squares, and Imaginary Roots, (as contradistinguished to what they call Real Roots, whether affirmative or Negative)... These *Imaginary* Quantities (as they are commonly called) arising from *Supposed* Root of a Negative Square, (when they happen) are reputed to imply that the Case proposed is Impossible.
The quotation above is from Chapter LXVI (p. 264), Of NEGATIVE SQUARES, and their IMAGINARY ROOTS in Algebra. This work is a translation of "De Algebra Tractatus; Historicus & Practicus" written in Latin in 1673. For the Latin edition of the latter consult "Opera mathematica", vol. II, Oxoniae, 1693. [Julio Gonz嫮ez Cabill鏮]

As a way of removing the stigma of the name, the American mathematician Arnold Dresden (1882-1954) suggested that imaginary numbers be called normal numbers, because the term "normal" is synonymous with perpendicular, and the y-axis is perpendicular to the x-axis (Kramer, p. 73).

The term IMAGINARY GEOMETRY was used by Lobachevsky, who in 1835 published a long article, "Voobrazhaemaya geometriya" (Imaginary Geometry).

The term IMAGINARY PART was used by Sir William Rowan Hamilton in an 1843 paper. He was referring to the vector and scalar portions of a quaternion [James A. Landau].

The term IMAGINARY UNIT was used (and apparently introduced) by by Sir William Rowan Hamilton in "On a new Species of Imaginary Quantities connected with a theory of Quaternions," Proceedings of the Royal Irish Academy, Nov. 13, 1843: "...the extended expression...which may be called an imaginary unit, because its modulus is = 1, and its square is negative unity."

IMPLICIT DEFINITION. In the literature of mathematics, this term was introduced by Joseph-Diaz Gergonne (1771-1859) in Essai sur la th廩rie des d嶨initions, Annales de Math幦atique Pure et Appliqu嶪 (1818) 1-35, p. 23. (The Annales begun to be published by Gergonne himself in 1810.) He also emphasized the contrast between this kind of definition and the other "ordinary" ones which, according to him, should be called "explicit definitions". According to his own example, given the words "triangle" and "quadrilateral" we can define (implicitly) the word "diagonal" (of a quadrilateral) in a satisfactory way just by means of a property that individualizes it (namely, that of dividing the quadrilateral in two equal triangles). Gergonne’s observations are now viewed by many as an anticipation of the "modern" idea of "definition by axioms" which was so fruitfully explored by Dedekind, Peano and Hilbert in the second half of the nineteenth century. In fact, still today the axioms of a theory are treated in many textbooks as "implicit definitions" of the primitive concepts involved. We can also view Gergonne’s ideas as anticipating, to a certain extent, the use of "contextual definitions" in Russell’s theory of descriptions (1905). [Carlos C廥ar de Arajo]

IMPROPER FRACTION was used in English in 1542 by Robert Recorde in The ground of artes, teachyng the worke and practise of arithmetike: "An Improper Fraction...that is to saye, a fraction in forme, which in dede is greater than a Unit."

IMPROPER DEFINITE INTEGRAL occurs in "Concerning Harnack's Theory of Improper Definite Integrals" by Eliakim Hastings Moore, Trans. Amer. Math. Soc., July 1901.

Improper integral appears in the same paper.

INCENTER is dated ca. 1890 in MWCD10.

The term INDEFINITE INTEGRAL is defined by Sylvestre-Fran蔞is Lacroix (1765-1843) in Trait du calcul diff廨entiel et integral (Cajori 1919, page 272).

INDEX. Schoner, writing his commentary on the work of Ramus, in 1586, used the word "index" where Stifel had used "exponent" (Smith vol. 2).

The term INDUCTION was first used in the phrase per modum inductionis by John Wallis in 1656 in Arithmetica Infinitorum. Wallis was the first person to designate a name for this process; Maurolico and Pascal used no term for it (Burton, page 440). [See also mathematical induction, complete induction, successive induction. ]

INFINITE DESCENT. Pierre de Fermat (1601-1665) used the term method of infinite descent (Burton, page 488; DSB).

The term INFINITELY SMALL was used by Christian Huygens (1629-1695) (DSB).

The term INFINITESIMAL ANALYSIS was used in 1748 by Leonhard Euler in Introductio in analysin infinitorum (Kline, page 324).

INFIX (notation) is found in D. Wood, "A proof of Hamblin's algorithm for translation of arithmetic expressions from infix to postfix form," BIT, Nordisk Tidskr. Inform.-Behandl. 9 (1969).

INFLECTION POINT appears in a 1684 paper by Leibniz, according to Katz (page 528), who has a footnote referring to Struik, Source Book, page 275.

INFORMATION THEORY. The OED2 shows a number of citations for this term from 1950.

INJECTION was used in 1950 by S. MacLane in the Bulletin of the American Mathematical Society (OED2).

INJECTIVE was used in 1952 by Eilenberg and Steenrod in Foundations of Algebraic Topology (OED2).

INNER PRODUCT. Schwartzman writes (p. 155):

When the German Sanskrit scholar Hermann Gnther Grassman (1809-1877) developed the general algebra of hypercomplex numbers, he realized that more than one type of multiplication is possible. To two of the many possible types he gave the names inner and outer. The names seem to have been chosen because they are antonyms rather than for any intrinsic meaning.
In English, inner product is found in a 1909 Webster dictionary, although Cajori (1928-29) uses the terms internal and external product.

The term INNUMERACY was popularized as the title of a recent book by John Allen Paulos. The word is found in the Random House Unabridged Dictionary, 2nd ed. (1987), and the word innumerate is dated 1959 in MWCD10.

INTEGER (as a noun) was used in English in 1571 by Thomas Digges (1546?-1595) in A geometrical practise named Pantometria. An earlier use exists for "integer number."

INTEGRABLE is found in English in 1727-41 in Chambers' Cyclopaedia (OED2).

The word INTEGRAL first appeared in print by Jacob (or James or Jacques I) Bernoulli (1654-1705) in May 1690 in Acta eruditorum, page 218. He wrote, "Ergo et horum Integralia aequantur" (Cajori vol. 2, page 182; Ball). According to the DSB this represents the first use of integral "in its present mathematical sense."

However, Jean I (or Johann or John) Bernoulli (1667-1748) also claimed to have introduced the term. According to Smith (vol. I, page 430), "the use of the term 'integral' in its technical sense in the calculus" is due to him.

The the following terms to classify solutions of nonlinear first order equations are due to Lagrange: complete solution or complete integral, general integral, particular case of the general integral, and singular integral (Kline, page 532).

INTEGRAL CALCULUS. Leibniz originally used the term calculus summatorius (the calculus of summation) in 1684 and 1686.

Johann Bernoulli introduced the term integral calculus.

Cajori (vol. 2, p. 181-182) says:

At one time Leibniz and Johann Bernoulli discussed in their letters both the name and the principal symbol of the integral calculus. Leibniz favored the name calculus summatorius and the long letter [long S symbol] as the symbol. Bernoulli favored the name calculus integralis and the capital letter I as the sign of integration. ... Leibniz and Johann Bernoulli finally reached a happy compromise, adopting Bernoulli's name "integral calculus," and Leibniz' symbol of integration.
According to Smith (vol. 2, page 696), Leibniz in 1696 adopted the term calculus integralis, already suggested by Jacques Bernoulli in 1690.

According to Stein and Barcellos (page 311), the term integral calculus is due to Leibniz.

The term "integral calculus" was used by Leo Tolstoy in Anna Karenina, in which a character says, "If they'd told me at college that other people would have understood the integral calculus, and I didn't, then ambition would have come in."

INTEGRAL DOMAIN is found in A. A. Albert, "Integral domains of rational generalized quaternion algebras," Bull. Am. Math. Soc. 40 (1934).

INTEGRAL EQUATION. According to Kline (page 1052) and Cajori 1919 (page 393), the term integral equation is due to Paul du Bois-Reymond (1831-1889), Jour. fr Math., 103, 1888, 288. However, Euler used a phrase which is translated integral equation in the paper "De integratione aequationis differentialis," Novi Commentarii Academiae Scientarum Petropolitanae 6, 1756-57 (1761) [James A. Landau].

The term INTEGRAL GEOMETRY is due to Wilhelm Blaschke (1885-1962), according to the University of St. Andrews website.

INTEGRAND was used in 1897 by H. F. Baker in Abel's Theorem: "The integrand of the Abelian integral u, is single-valued on the Riemann surface" (OED2).

INTEGRATING FACTOR occurs in "Three Particular Systems of Lines on a Surface," Luther Pfahler Eisenhart, Transactions of the American Mathematical Society, Vol. 5, No. 4. (Oct., 1904).

The term INTEGRATION BY PARTS appears in the 1902 Encyclopaedia Britannica in the article "Infinitestimal Calculus" and also appears in the first edition of the OED.

The method was invented by Brook Taylor and discussed in Methodus incrementorum directa et inversa (1715).

INTERMEDIATE VALUE THEOREM appears in 1937 in Differential and Integral Calculus, 2nd. ed. by R. Courant [James A. Landau].

The term INTERPOLATION was introduced into mathematics by John Wallis (DSB; Kline, page 440).

INTERSECTION (in set theory) is found in Webster's New International Dictionary of 1909.

INTRINSICALLY CONVERGENT SEQUENCE is the term used by Courant for "Cauchy sequence" in Differential and Integral Calculus, 2nd. ed. (1937) [James A. Landau].

The term INVARIANT is due to James Joseph Sylvester (1814-1897), according to Cajori (1919, page 345) and Kline (page 927), who supplies the reference Coll. Math. Papers, I, 273.

According to Kramer (p. 388), Galois used the adjective "invariant" referring to a normal subgroup.

INVERSE (as a noun, the element which produces the identity element) appears in 1900 in Ann. Math. (OED2).

The term INVERSE GAUSSIAN DISTRIBUTION was coined in 1948 by M. C. K. Tweedie, according to Gerard Letac.

INVERSE VARIATION is found in 1881 in Elements of Algebra by G. A. Wentworth [James A. Landau].

INVERTIBLE is found in the phrase "invertible elements of a monoid A" in 1956 in Fundamental Concepts of Algebra ii. 27 by C. Chevelley (OED2).

The term INVOLUTION is due to G廨ard Desargues (1593-1662) (Kline, page 292).

IRRATIONAL is used in English by Robert Recorde in 1551 in The Pathwaie to Knowledge: "Numbres and quantitees surde or irrationall."

The term IRREDUCIBLE INVARIANT was used by Arthur Cayley (1821-1895).

ISOGRAPHIC is the word used by Ernest Jean Philippe Fauquede Jonqui廨es (1820-1901) to describe the transformations he had discovered, later called birational transformations (DSB).

ISOMETRIC is found in English in an 1855 dictionary. Isometrische Abbildung (isometric mapping) is found in the 1944 edition of Hausdorff's Grundzuge der Mengenlehre and may occur in the first 1914 edition [Gerald A. Edgar].

ISOMETRY (equality of measure) is found in English in a 1909 Webster dictionary. Aristotle used the word isometria.

In its modern sense, isometry occurs in English in 1941 in Survey of Modern Algebra by MacLane and Birkhoff: "An obvious example is furnished by the symmetries of the cube. Geometrically speaking, these are the one-one transformations which preserve distances on the cube. They are known as 'isometries,' and are 48 in number" (OED).

ISOMORPHISM was used by Walter Dyck (1856-1934) in 1882 in Gruppentheoretische Studien (Katz, page 675).

ISOSCELES was used in English 1551 by Robert Recorde in The Pathwaie to Knowledge: "There is also an other distinction of the name of triangles, according to their sides, whiche other be all equal...other els two sydes bee equall and the thyrd vnequall, which the Greekes call Isosceles, the Latine men aequicurio, and in english tweyleke may they be called."

In English, an isosceles triangle was called an equicrure in 1644 and an equicrural triangle in 1650 (OED2). These are the earliest uses for the alternate term of Latin origin in the OED2.

The term ITERATED FUNCTION SYSTEM was coined by Michael Barnsley, according to an Internet website.


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