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Earliest Known Uses of Some of the Words of Mathematics (I)

Last revision: Aug. 27, 2018

ICOSAHEDRON. The icosahedron is treated inEuclid’s ElementsXIII. xvi.The word is formed from twenty + seat, base.The English word is found in Sir Henry Billingsley’s 1570 translation of theElements (OED).

IDEAL (point or line) was introduced asidéal byJ. V. Poncelet inTraité des Propriétés proj.des Figures (1822).

IDEAL (number theory) was introduced by Richard Dedekind (1831-1916) in his edition ofP. G. L.DirichletVorlesungen über Zahlentheorie (ed. 2, 1871) Suppl. x. p. 452. According to Kline (p. 823), the name waschosen “in honor of Kummer’s ideal numbers.” See also the article in theEncyclopaedia of Mathematics.

The term passed without change into English. AJSTOR search produced references toDedekind’s theory of ideals in Arthur S. Hathaway “A Memoir in the Theoryof Numbers,”AmericanJournal of Mathematics,9,(Jan., 1887), p. 166: “The theory of ideals here established, however, differsfrom Dedekind’s theory in important respects…” [John Aldrich]

See the entryIDEAL NUMBER.

IDEAL NUMBER. According to Kline (p. 819), Ernst Eduard Kummer (1810-1893) createda theory of ideal numbers in the papers he published in theJournalfür die reine und angewandte Mathematik,12, 1847: “ZurTheorie der complexen Zahlen” (pp. 319-326) and “Ueber die Zerlegung der ausWurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren” (pp.327-367). See also the article in theEncyclopaedia of Mathematics.

The English term is found in H. J. S. Smith’s “Reporton the Theory of Numbers.—Part II”Report OfThe Thirty-First Meeting Of The British Association For The Advancement OfScience; Held At Manchester In September 1861. (The text isavailable through Google books.) [John Aldrich]

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

When an expression raised to the square or any higherpower vanishes, it may be callednilpotent; but when, raisedto a square or higher power, it gives itself as the result, it may becalledidempotent.

The defining equation of nilpotent and idempotentexpressions are respectivelyAⁿ = 0, andAⁿ =A; but with reference to idempotentexpressions, it will always be assumed that they are of theform

unless it be otherwise distinctly stated.

The OED2 shows a 1937 citation with a simplified definition ofidempotent inModern Higher Algebra (1938) iii 88 by A.A. Albert: "A matrixE is called idempotent ifE² =E. [Older dictionaries pronounceidempotent with the only stress on the second syllable, but newerones show a primary stress on the first syllable and a secondarystress on the penult.]

The termIDENTIFIABILITY was coined bythe econometrician Tjalling C. Koopmans around 1945 with reference to theeconomicidentity of a relationship within asystem of relationships. (SeeSIMULTANEOUS EQUATIONS MODEL). The term began appearing in the econometrics literatureimmediately, although Koopmans’s own exposition of the subject did not appearuntil 1949--his "Identification Problems in Economic ModelConstruction"Econometrica,17, pp. 125-144. Around 1950 the term wastaken up by statistical theorists and used in a more general sense, see e.g.Jerzy Neyman’s "Existence of Consistent Estimates of the DirectionalParameter in a Linear Structural Relation Between Two Variables,"Annals of Mathematical Statistics,22, (1951), pp. 497-512. [John Aldrich]

IDENTITY (type of equation) is found in 1831 in the secondedition ofElements of the Differential Calculus (1836) byJohn Radford Young: "This is obvious, for this first term is what thewhole development reduces to whenh = 0, but we must in thiscase have the identityf(x) =f(x); hencef(x) is the first term" [James A. Landau].

Young also uses the termidentical equations in the same work.

IDENTITY andIDENTITY-ELEMENT are found in 1894 in“The Group of Holoedric Transformation Into Itself of a GivenGroup” by Prof. E. Hastings Moore inBulletin of the American Mathematical Society. [OEDand Google print search by James A. Landau]

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

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

An earlier term wasunit-matrix; see e.g. J.J. Sylvester "Note on the Theory of Simultaneous Linear Differentialof Difference Equations with Constant Coefficients,"American Journal ofMathematics,4, (1881), 321-326,Coll Math Papers, III, pp. 551-6.Thematrix unity was the term used by A. Cayley "A Memoir on the Theoryof Matrices" (1858)Coll Math Papers, II, p. 477.

IFF. The earliest citation for “iff” in theOED is from John L. Kelley’sGeneral Topology (1955): “F isequicontinuous atx iff there is a neighborhood ofx whose imageunder every member of F is small.” Kelley credited the term toPaul R. Halmos.

On the last page of his autobiography, Halmos writes:

My most nearly immortal contributions are an abbreviation and a typographical symbol. Iinvented “iff”, for “if and only if”—but I could never believe that I wasreally its first inventor. I am quite prepared to believe that it existedbefore me, but I don'tknow thatit did, and my invention (re-invention?) of it is what spread it through themathematical world. The symbol is definitely not my invention—it appeared inpopular magazines (not mathematical ones) before I adopted it, but, once again,I seem to have introduced it into mathematics. It is the symbol that sometimeslooks like▐, and is used to indicate an end, usually the endof a proof. It is most frequently called the “tombstone”, but at least onegenerous author referred to it as the “halmos”.

FromI Want to Be aMathematician: An Automathography, by Paul R. Halmos, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985, page 403. SeeWolfram: iff.

For the halmos, seeEarliest Uses of Symbols of Set Theory and Logic.

The termsIMAGINARY andREAL were introduced in Frenchby Rene Descartes (1596-1650) in "La Geometrie" (1637):

...tant les vrayes racines que les fausses ne sont pastousiours réelles; mais quelquefois seulement imaginaires;c'est à dire qu'on peut bien toujiours en imaginer autant que aiydit en chàsque Equation; mais qu'il n'y a quelquefois aucunequantité, qui corresponde à celles qu'on imagine. commeencore qu'on en puisse imaginer trois en celle cy, x³ - 6xx+ 13x - 10 = 0, il n'y en a toutefois qu'une réelle, qui est 2,& pour les deux autres, quois qu'on les augmente, ou diminué,ou multiplié en la façon que ie viens d'éxpliquer,on ne sçauroit les rendre autres qu'imaginaires. [...neither thetrue roots nor the false are always real; sometimes they are, however,imaginary; namely, whereas we can always imagine as many roots foreach equation as I have predicted, there is still not always aquantity which corresponds to each root so imagined. Thus, while wemay think of the equation x³ - 6xx + 13x - 10 = 0 as havingthree roots, yet there is just one real root, which is 2, and theother two, however, increased, diminished, or multiplied them as wejust laid down, remain always imaginary.] (page 380)

Imaginary andreal are found in English in 1668 inPhilosophical Transactions. The words are found in a review ofthe bookGeometriæ Universalis: “And for the likereason aCubick Æquation, having three reals roots, cannever be reduced to apure Æquation, which hath but oneonely root, for in these Æquations,Reduction shall no wiseprofit, forasmuch as ‘tis impossible, by aid thereof to change anImaginary root into a real one, and the Converse.” [Google printsearch by James A. Landau]

As a way of removing the stigma of the name, the Americanmathematician Arnold Dresden (1882-1954) suggested that imaginarynumbers be callednormal numbers, because the term "normal" issynonymous with perpendicular, and the y-axis is perpendicular to thex-axis (Kramer, p. 73). The suggestion appears in 1936 in hisAnInvitation to Mathematics.

Some other terms that have been used to refer to imaginary numbersinclude "sophistic" (Cardan), "nonsense" (Napier), "inexplicable"(Girard), "incomprehensible" (Huygens), and "impossible" (manyauthors).

The first edition of theEncyclopaedia Britannica (1768-1771) has:"Thus the square root of -a² cannot be assigned, and iswhat we call animpossible orimaginary quantity."

There are two modern meanings of the termimaginary number.InMerriam-Webster’s Collegiate Dictionary, 10th ed., animaginary number is a number of the forma +bi whereb is not equal to 0. InCalculus and Analytic Geomtry(1992) by Stein and Barcellos, "a complex number that lies onthey axis is calledimaginary."

The termIMAGINARY GEOMETRY was used by Lobachevsky, who in1835 published a long article, "Voobrazhaemaya geometriya" (ImaginaryGeometry).

The termIMAGINARY PART appears in 1748inA Treatise of Algebra: In Three Partsby Colin Maclaurin. [Google print search by James A. Landau]

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

IMPLICIT DEFINITION. In the literature of mathematics, thisterm was introduced by Joseph-Diaz Gergonne (1771-1859) inEssaisur la théorie des définitions, Annales deMathématique Pure et Appliquée (1818) 1-35, p. 23. (TheAnnales begun to be published by Gergonne himself in 1810.) Healso emphasized the contrast between this kind of definition and theother "ordinary" ones which, according to him, should becalled "explicit definitions". According to his ownexample, given the words "triangle" and"quadrilateral" we can define (implicitly) the word"diagonal" (of a quadrilateral) in a satisfactory way justby means of aproperty that individualizes it (namely, that ofdividing the quadrilateral in two equal triangles). Gergonne’sobservations are now viewed by many as an anticipation of the"modern" idea of "definition by axioms" which wasso fruitfully explored by Dedekind, Peano and Hilbert in the secondhalf of the nineteenth century. In fact, still today the axioms of atheory are treated in many textbooks as "implicitdefinitions" of the primitive concepts involved. We can alsoview Gergonne’s ideas as anticipating, to a certain extent, theuse of "contextual definitions" in Russell’s theoryof descriptions (1905). [Carlos César de Araújo]

IMPLICIT DIFFERENTIATION is found in English in 1836 inThe differential and integral calculusby Augustus De Morgan.[Google print search]

IMPLICIT FUNCTION is found in 1814New Mathematical andPhilosophical Dictionary: "Having given the methods ... ofobtaining the derived functions, of functions of one or morequantities, whether those functions be explicit or implicit, ... wewill now show how this theory may be applied" (OED2).

IMPROPER FRACTION was used in English in 1542 by RobertRecorde inThe ground of artes, teachyng the worke and practise ofarithmetike: "An Improper Fraction...that is to saye, a fractionin forme, which in dede is greater than a Unit."

IMPROPER INTEGRAL occurs in "ConcerningHarnack’s Theory of Improper Definite Integrals" by EliakimHastings Moore,Trans. Amer. Math. Soc., July 1901.The term may be much older.

INCENTER, INCIRCLE. See the entryCIRCUMCENTER, CIRCUMCIRCLE, EXCENTER, EXCIRCLE, INCENTER, INCIRCLE, MIDCIRCLE.

INCLUDED ANGLE andINCLUDED SIDE are found in 1753 inThe Elements of Algebra in a New and Easy Method: Second Edition, Corrected,by Nathaniel Hammond. [Google print search by James A. Landau]

INCOMMENSURABLE.Incommensurability is found in Latinin the 1350s in the titleDe commensurabilitate siveincommensurabilitate motuum celi (the commensurability orincommensurability of celestial motions) by Nicole Oresme.

The termINDEFINITE INTEGRAL is defined bySylvestre-François Lacroix (1765-1843) inTraité ducalcul différentiel et integral (Cajori 1919, page 272).

Indefinite integral appears in English in1823 inDictionary of the Mathematical and Physical Sciences, According to the Latest Improvements and Discoveries,Edited by James Mitchell. [Google print search by James A. Landau]

INDEPENDENT EVENT andDEPENDENT EVENT are found in 1738inThe Doctrine of Chances by De Moivre: "Two Events areindependent, when they have no connexion one with the other, and thatthe happening of one neither forwards nor obstructs the happening ofthe other. Two events are dependent, when they are so connectedtogether as that the Probability of either’s happening is alter'd bythe happening of the other."

INDEPENDENT VARIABLE. SeeDEPENDENT VARIABLE.

The termINDETERMINATE FORM is used in French in 1840 inMoigno, abbé (François Napoléon Marie),(1804-1884):Leçons de calcul différentiel et de calculintégral, rédigées d'après lesméthodes et les ouvrages publiés ou inédits de M.A.-L. Cauchy, par M. l'abbé Moigno. [V. Frederick Rickey]

Indeterminate Forms is found in English as a chapter title in1841 inAn Elementary Treatise on Curves, Functions, andForces by Benjamin Pierce. [Google print search by James A. Landau]

Forms such as 0/0 are calledsingular values andsingularforms in in 1849 inAn Introduction to the Differential andIntegral Calculus, 2nd ed., by James Thomson.

InPrimary Elements of Algebra for Common Schools andAcademies (1866) by Joseph Ray, 0/0 is called "the symbol ofindetermination."

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

INDEX NUMBER became a standard term in economic statistics in the 1880s. The OED’s earliest quotation is from 1875William Stanley JevonsMoney and the Mechanism of Exchangechapter xxv p. 332:"In the annual Commercial History and Review of theEconomist newspaper, there has, for many years,appeared a table containing the Total Index Number of prices, or the arithmeticalsum of the numbers expressing the ratios of the prices of many commodities tothe average prices of the same commodities in the years 1845-50." The table was the work ofWilliam Newmarch butJevons had himself pioneered the method of index numbers inA Serious Fallin the Value of Gold Ascertained and its Social Effects Set Forth (1863).Anticipations of the technique have been found in the early 18^thcentury, in the work ofWilliamFleetwood. See M. G. Kendall "The Early History of Index Numbers"in M. G. Kendall & R. L. PlackettStudies in the History of Statisticsand Probability, volume 2 (1977).

INDICATOR. Seetotient.

INDICATOR FUNCTION andINDICATOR RANDOM VARIABLE.The termindicator of a set appears in M. Loève’sProbability Theory(1955) and, according to W. Feller (An Introduction to Probability Theory and its Applications volumeII), Loève was responsible for the term. Loève’sProbability Theorydid not use termindicator random variable but this soon appeared, see e.g. H. D. Brunk’s"On an Extension of the Concept Conditional Expectation"Proceedings of the American MathematicalSociety,14, (1963), pp. 298-304.

SeeCHARACTERISTIC FUNCTION of a set andBERNOULLI DISTRIBUTION.

INDUCE. Adolf Hurwitz introduced the use of the verbinduce in mathematics,as in the case in which a rotation of the octahedron induces a permutation of its vertices,according to the third edition ofDie Theorie der Gruppen von endlicher Ordnung by Andreas Speiser.[Information provided by Robert Alan Wolf.]

The termINDUCTION was first used in the phraseper moduminductionis by John Wallis in 1656 inArithmeticaInfinitorum. Wallis was the first person to designate a name forthis process; Maurolico and Pascal used no term for it (Burton, page440). [See alsomathematical induction, complete induction,successive induction. ]

INDUCTIVE (PARTIALLY) ORDERED SET. The adjective "inductive"used in this context was introduced by Bourbaki inÉlementsde mathématique. I. Théorie des ensembles. Fascicule derésultats, Actualités Scientifiques etIndustrielles, no. 846, Hermann, Paris, 1939. Bourbaki’s originalterm and definition is now standard among mathematicians: a poset(X, ) is inductive if everytotally ordered subset of it has a supremum, that is:

(1) (AX) (A is a chainA has a supremum).

A notion of "completeness" is usually associated with conditions ofthis kind. Thus, if

(2) (AX) (A has a supremum),

then (X, ) becomes a"complete lattice." Similarly, (X, ) is said to be "order-complete" (or"Dedekind-complete") if

(3) (AX) (A andA has an upper boundA has a supremum).

This may explain why some computer scientists prefer the term"complete poset" instead of "inductive poset." (However, "completeposet" is also used by many of them in a related but differentsense.)

[Carlos César de Araújo]

INFINITE DESCENT. Pierre de Fermat (1607?-1665) used the termmethod of infinite descent (Burton, page 488; DSB).

A paper by Fermat is titled "La méthode de la 'descenteinfinie ou indéfinie.'" Fermat stated that he named themethod.

INFINITELY SMALL. A Google print search shows the Latin terminfinite parva used by John Wallis in 1655 and 1656 [Mikhail Katz].

Infinitely small is found in English in 1685 inA Treatise of Algebra Both Historical and Practical by Jon Wallis.

According to theDSB, Christian Huygens used the terminfinitely small.

INFINITESIMAL. In a letter from Wallis to the French mathematician Vincent Léotaud, dating from February 17/[27], 1667/[1668], a few months before the publication of Mercator’sLogarithmotechnia,Wallis proposes a change of terminology to Léotaud:

Dic Angulum Contactûs esse, Infinitesimam partem duorum Rectorum: (seu 2/∞ R.) [Say the angle of contact is an infinitesimal part [pars infinitesima]of two right angles (or 2/∞ R.)]

Wallis usedinfinitesima in print in 1670:

Linea, ex infinitis punctis, hoc est, Lineolis infinite exiguis, longitudine aequalibus, vel aeque altis; quarum cujusvis longitudo vel altitudo sit 1/∞ (pars infinitesima) longitudinis vel altitudinis totius lineae

(Wallis,Mechanica, p. 110).

Leibniz seems to have started to use the terminfinitesimalin late spring 1673 (see Gottfried Wilhelm Leibniz,Sämtliche Schriften und Briefe, Reihe VII: Mathematische Schriften, vol 4: Infinitesimalmathematik 1670-1673,Berlin : Akademie Verlag, 2008, pp. 263, 356, 398 etc.)on line.

Apparently Leibniz took the term from Nicholas Mercator’swritings, as he was to recall much later in a letter to John Wallis,March 30/[April 9], 1699 (Leibniz, Mathematische Schriften (ed.Gerhardt), vol. IV, p. 63). In his bookLogarithmotechnia (1668) Mercator does not use the terminfinitesimal, but insteadpars infinitissima (pp. 30–34). But at the end of his article“Some illustration of the Logarithmotechnia,”Philosophical Transactions n° 38(1668), pp. 759-764, Mercator added a note:“atque ubicunque, Lector offenderit infinitissimam, legatinfinitesimam.” This was a reaction to the critique by Wallis atthe beginningof his review of the book, printed in the same issue of thePhilosophical Transactions, pp. 753-759.

[This information was contributed by Dr. Siegmund Probst.]

Infinitesimal is found in English with its modern meaning in 1706inAn Institution of Fluxions by Humphrey Ditton.

The word was used by Tolstoy inWar and Peace: “Arriving at infinitesimals, mathematics,the most exact of sciences, abandons the process of analysis and enters on the new process ofthe integration of unknown, infinitely small, quantities.”

The termINFINITESIMAL ANALYSIS was used in 1748 by LeonhardEuler inIntroductio in analysin infinitorum (Kline, page 324).

The termINFINITESIMAL CALCULUS is found in English in 1797 in the 3rd edition oftheEncyclopaedia Britannica. [Google print search by James A. Landau]

INFIX NOTATION is found in 1962 inAlgebraic Logic by PaulHalmos. [Google print search by James A. Landau]

INFLECTION POINT. In the last page of an appendix to his"Methodus ad disquirendam maximam & minimam," Fermat wrote:

Quia tamen sæpius curvatura mutatur, ut in conchoide Nicomedea, quæ pertinetad priorem casum, et in omnibus speciebus curvæ Domini de Roberval (prima excepta) quæ pertinetad secundum, ut perfecte curva possit delineari, investiganda sunt ex arte puncta inflexionum,in quibus curvatura ex convexa fit concava vel contra: cui negotio eleganter inservit doctrinade maximis et minimis, hoc prwmisso lemmate generali:

The above was posthumously printed in theOpera varia, Toulouse, 1679, and is also inOeuvresde Fermat, Paris : Gauthier-Villars, 1891, p. 166).

The inflection point of this curve had been mentioned by Roberval in his letter to Fermat from22 November, 1636 (printed in Oeuvres de Fermat - Supplément aux tomes I-IV, Paris, 1922, p. 44-45);Roberval writes concerning the conchoid:

vous treuverez qu'il y a un certain point en icelle comme B,tel que depuis A iusques en B, elle est convexe en dehors, et depuis B par C à l'infiny, elle estconvexe en dedans, ce qui est admirable. Et encor plus que par le point B, il ne se peut mener deligne droite qui la touche, mais une comme OBY, qui fera les angles au sommet OBA, YBC, chacun moindrequ'aucun angle rectiligne donne. Il en sera de mesme de l'autre part, et ce sont ces deux points d'oùie vous mandois qu'on ne pouvoit mener de tangentes. (p. 45)

Mesolabum, (2nd ed., 1668) by René François Walther de Sluze contains two additional parts, "De Analysi" and "Miscellanea."Chapter 5 of the Miscellanea, which has the title "De puncto flexus contrarii, in Conchoeide Nicomedis prima,"answers a problem posed (and solved) by Christiaan Huygens (mentioned p. 119) at the end of his"De circuli magnitudine inventa," Leiden, 1654: problem VIII "In Conchoide linea invenire confinia flexus contrarii,"p. 69-71 (with French translation and commentary in Huygens,Oeuvres complètes,vol. XII, 1910, p. 210-215; p. 110-112give more detailed information on the interchange between Huygens andSchooten concerning this problem). The problem was later also solved byHendrik van Heuraet, printed in Frans van Schooten’s"In Geometriam Renati Des Cartes Commentarii," 2nd edition, 1659, p.258-262,where Schooten calls the point "punctum Conchoïdis C, quod duasejus portiones, concavam & convexam, à se invicemdistinguit."

The review of Sluse’s book inPhilosophical Transactions, nr 45, (1669), p. 903-909, treats chapter 5on p. 907 and quotes the title, but does not translate the term into English.

In the extract of Sluse’s famous letter on his tangent method, printed inPhilosophical Transactions,nr 90, (1673), p. 5143-5147,Sluse refers to chapter 5 of the "Miscellanea": "qua ratione flexuscontrarii curvarum ex Tangentibus inveniantur, ostendi" (p. 5147).

In his 1684 article "Nova methodus pro maximis et minimis" inActa eruditorum, Leibniz usedpunctum flexus contrarii (point of opposite flection).More information is in T. F. Mulcrone,The Mathematics Teacher 61 (1968), 475-478,which has not been seen.

Newton used the term "point of straightness" to describe inflection points, according to the web pageThe History of Curvature.

Point of Inflection andpoint of inflexion are found in English in 1702 inA mathematical dictionary by Joseph Raphson, abridged from M. Ozanam and others.Page 16 has “point of Inflection of a Curve line” and page 17 has “Point of Inflexion of a Curve.”

InAn Elementary Treatise on Curves, Functions and Forces(1846), Benjamin Peirce writes, "When a curve is continuous at apoint, but changes its direction so as to turn its curvature theopposite way at this point, the point is calleda point ofcontrary flexure, ora point of inflexion."

This entry was largely contributed by Siegmund Probst, who found the material on Fermat and Robervalin Cantor’sVorlesungen über Geschichte der Mathematik.

INFLUENCE CURVE (in engineering) is found in 1902 in the article on “Bridges”in the 10th ed. of theEncyclopaedia Britannica. [Google print search by James A. Landau]

INFLUENCE CURVE (in robust statistical inference). The earliestJSTORappearance of the term is in Peter J. Huber "The 1972 Wald Lecture Robust Statistics: A Review,"Annals of Mathematical Statistics,43,(1972), 1041-1067. Huber attributes the term to Frank R. Hampel. A little laterHampel’s "The Influence Curve and Its Role in Robust Estimation,"Journal of the American Statistical Association,69, (1974). 383-393 appeared.

INFORMATION, AMOUNT OF, QUANTITY OF in the theory of statistical estimation.R. A. Fisher first wroteabout "the whole of the information which a sample provides" in 1920 (Mon. Not. Roy. Ast. Soc., 80, 769). In 1922-5 he developed the idea that information could be given quantitative expression as minus the expected value of the second derivative of the log-likelihood. The formula for "the amount of information in a single observation" appears in the 1925 "Theory of Statistical Estimation.,"Proc. Cambr. Philos. Soc. 22. p. 709. In the modern literature thequalification Fisher’s information is common, distinguishing Fisher’smeasure from others originating in the theory of communication aswell as in statistics. [John Aldrich and David (1995)].

INFORMATION, MEASURE OF in the theory of communication was introduced byClaude Shannon in hisA Mathematical Theory of Communication(1948).

INFORMATION THEORY. The OED2 shows a number of citations for this term from 1950: e.g. W. G. Tuller inTrans. Amer. Inst. Electr. Engin. LXIX. 1612/1 "The statistical theory of communications, developed over the past few years and often called information theory, can be of real assistance in the design of communication systems." The theory alluded to was mainly the work ofC. E. Shannon,in particular hisA Mathematical Theory of Communication (1948).

See alsoBIT andENTROPY.

INJECTION, SURJECTION andBIJECTION. TheOED records a use ofinjectionby S. MacLane in theBulletin of the American Mathematical Society (1950) andinjective in Eilenbergand Steenrod inFoundations of AlgebraicTopology (1952). However the family of terms is introduced on p. 80 ofNicholas Bourbaki’sThéorie des ensembles, Éléments demathématique Première Partie, Livre I, Chapitres I, II (1954). Reviewing the book in theJournal of Symbolic Logic, R. O. Gandy (1959, p. 72) wrote:

Another usefulfunction of Bourbaki’s treatise has been to standardise notation andterminology… Standard terms are badly needed for “one-to-one,” “onto” and“one-to-one onto”; will Bourbaki’s “injection,”“surjection” and “bijection” prove acceptable?

The termsdid prove acceptable, even tomathematicians writing in English, and quickly became standard. For instance,all three terms are used in Jun-Ichi Igusa “Fibre Systems of JacobianVarieties,”American Journal of Mathematics,78, (1956), 171-199.(JSTOR search) The adjectival forms appear in C. Chevalley,FundamentalConcepts of Algebra (1956): “A homomorphism which is injective is called amonomorphism; a homomorphism which is surjective is called an epimorphism.” (OED)

See the entriesONE-TO-ONE andONTO andMONOMORPHISM.

The termINNER PRODUCT was coined (in German asinneresprodukt) by Hermann Günther Grassman (1809-1877) inDielineale Ausdehnungslehre (1844).

According to the OED2 it is "so named because an inner product of twovectors is zero unless one has a component 'within' the other, i.e.in its direction."

According to Schwartzman (p. 155):

When the German Sanskrit scholar Hermann GüntherGrassman (1809-1877) developed the general algebra of hypercomplexnumbers, he realized that more than one type of multiplication ispossible. To two of the many possible types he gave the namesinner andouter. The names seem to have been chosenbecause they are antonyms rather than for any intrinsicmeaning.

In English,inner product is found in 1901 inAn Elementary Exposition of Grassmann’s Ausdehnungslehre, or Theory of Extension byJos. V. Collins, reprinted from the American Mathematical Monthly, Vols. VI and VII. [Googleprint search by James A. Landau]

The termINNUMERACY was popularized as the title of a recentbook by John Allen Paulos. The word is found in 1959 inRep.Cent. Advisory Council for Educ. (Eng.) (Ministry of Educ.): "Ifhis numeracy has stopped short at the usual Fifth Form level, he isin danger of relapsing into innumeracy" (OED2).

INSTRUMENTAL VARIABLES in Econometrics and Statistics.The term first appeared in Olav Reiersøl’s "Confluence Analysis by Means of InstrumentalSets of Variables,"Arkiv for Mathematik, Astronomi och Fysik,32 (1945), 1-119.In anET interview (p.119) Reiersøl recalled that the name was suggested by hissupervisor, Ragnar Frisch. Reiersølhad proposed the method already in 1941 and independently R. C. Geary proposedit in 1943. Reiersøl and Geary were interested in errors in variables modelsbut around 1950 it became clear that their method could be applied to the recentlydevelopedSIMULTANEOUS EQUATIONS MODEL. Curiously the method had  been usedin the 1920s by the economist Phillip G. Wright and his geneticist son SewallWright for estimating demand functions but the method was not taken up and thework had no influence in Econometrics. The relative contributionof father and son has been a matter of controversy and Stock & Trebbi havediscussed the issue quite recently.

(Based on J. Aldrich "Reiersøl, Geary and the Idea of Instrumental Variables,"Economic and Social Review,24, (1993), 247-274pdf andJ. H. Stock & F Trebbi "Who Invented IV Regression?"Journal of Economic Perspectives,17, (2003), 177-194pdf)

See the entriesERROR: ERRORS IN VARIABLES andPATH ANALYSIS.

INTEGER andWHOLE NUMBER. Writing in Latin, Fibonacciusednumerus sanus.

According to Heinz Lueneburg, the termnumero sano "was usedextensively by Luca Pacioli in hisSumma. Before Pacioli, itwas already used by Piero della Francesca in hisTrattatod'abaco. I also find it in the second edition of Pietro Cataneo’sLe pratiche delle due prime matematiche of 1567. I haven'tseen the first edition. Counting also Fibonacci’s Latinnumerussanus, the wordsano was used for at least 350 years todenote an integral (untouched, virginal) number. Besides the wordssanus, sano, the wordsinteger, intero, intiero werealso used during that time."

The first citation forwhole number in the OED2 isfrom about 1430 inArt of Nombryng ix. EETS 1922:

Of nombres one is lyneal, ano(th)er superficialle,ano(th)er quadrat, ano(th)cubike or hoole.

Whole number is found in its modern sense in the title of oneof the earliest and most popular arithmetics in the English language,which appeared in 1537 at St. Albans. The work is anonymous, and itslong title runs as follows: "An Introduction for to lerne to rekenwith the Pen and with the Counters, after the true cast of arismetykeor awgrym in hole numbers, and also in broken" (Julio GonzálezCabillón).

Oresme usedintégral.

Integer was used as a noun in English in 1571 by Thomas Digges(1546?-1595) in A geometrical practise named Pantometria:"The containing circles Semidimetient being very nighe 11 19/21 forexactly nether by integer nor fraction it can be expressed" (OED2).

Integral number appears in 1658 in Phillips: "In Arithmetickintegral numbers are opposed to fraction[s]" (OED2).

Whole number is most frequently defined as Z+, although it issometimes defined as Z. InElements of the Integral Calculus(1839) by J. R. Young, the author refers to "a whole number or 0" butlater refers to "a positive whole number."

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

Integrable is also found in 1734 inAn Examination of Dr. Burnet’s Theory of the Earthby John Keill and Maupertuis. [Google print search]

The wordINTEGRAL first appeared in print by Jacob Bernoulli (1654-1705) in May 1690 inActaeruditorum, page 218. He wrote, "Ergo et horum Integraliaaequantur" (Cajori vol. 2, page 182; Ball). According to the DSBthis represents the first use ofintegral "in its presentmathematical sense."

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

The following terms to classify solutions of nonlinear first orderequations are due to Lagrange:complete solution orcomplete integral, general integral, particular case of thegeneral integral, andsingular integral (Kline, page 532).

INTEGRAL CALCULUS. Leibniz originally used the termcalculus summatorius (the calculus of summation) in 1684 and1686.

Johann Bernoulli introduced the termintegral calculus.

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

At one time Leibniz and Johann Bernoulli discussed intheir letters both the name and the principal symbol of the integralcalculus. Leibniz favored the namecalculus summatorius andthe long letter [long S symbol] as the symbol. Bernoulli favored thenamecalculus integralis and the capital letterI asthe sign of integration. ... Leibniz and Johann Bernoulli finallyreached a happy compromise, adopting Bernoulli’s name "integralcalculus," and Leibniz' symbol of integration.

According to Stein and Barcellos (page 311), the termintegralcalculus is due to Leibniz.

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

INTEGRAL DOMAIN is found in 1907 inSynopsis of Linear Associative Algebra byJames Byrnie Shaw. [Google print search by James A. Landau]

INTEGRAL EQUATION. According to Kline (p. 1052) and J. DieudonnéHistory of Functional Analysis (p. 97), the termintegralequation (Integralgleichung)is due to Paul du Bois-Reymond (1831-1889). Du Bois-Reymond introduced it inhis paper on theDIRICHLETPROBLEM, “Bemerkungen über,”Journalfür die reine und angewandte Mathematik,103, (1888), 204-229.Integral equations became a very hot topic at the beginning of the 20^th century with influential contributions from E. Fredholm,“Sur une classe des équations fonctionnelles,”Acta Mathematica,27,(1903), 365–390, and from D. Hilbert whose articles of 1904-6 werecollected into a book,Grundzüge einerallgemeinen Theorie der linearen Integralgleichungen (1912).Hilbert’s work played a critical part in the rise ofFUNCTIONAL ANALYSIS.See theEncyclopaedia of Mathematics entryIntegral equations.

Although du Bois-Reymond introduced the term in itsmodern sense, Euler had earlier used a phrase which is translatedintegral equation in the paper “Deintegratione aequationis differentialis,”Novi Commentarii Academiae Scientarum Petropolitanae 6,1756-57 (1761).Integral equation is found in English in 1802 in Woodhouse,Phil.Trans. XCII. 95: “Expressions deduced from the true integralequations.” (OED).

[This entry was contributed by John Aldrich. James A. Landau provided a citation.]

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

INTEGRAND (in logic). Sir William Hamilton of Scotland used this word in logic. It appears in hisLectures on metaphysics and logic (1859-1863):"This inference of Subcontrariety I would call Integration, because the mind here tends to determineall the parts of a whole, whereof a part only has been given. The two propositions together might be calledthe integral or integrant (propositiones integrales vel integrantes). The given proposition would bestyled the integrand (propositio integranda); and the product, the integrate (propositio integrata)"[University of Michigan Digital Library].

INTEGRAND (in calculus) is found in 1827 inBündige und reine Darstellung des wahrhaften Infinitesimal-Calculsby Friedrich Gottlieb von Busse. [Google print search by James A. Landau]

INTEGRATING FACTOR is found in 1832 inA Treatise on Differential Equations byW. C. Ottley. [Google print search by James A. Landau]

INTEGRATION BY PARTS is found in 1804 inTransactionsof the Cambridge Philosophical Society.[Google print search by James A. Landau]

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

INTEGRATION BY SUBSTITUTION is found in 1837 inThe Theoryof the Differential and Integral Calculus by John Forbes: “Thegeneral principle of the Integral Calculus is extended by means of thesedifferent modes, to the integration of complex functions, both rationaland surd; viz. integration by substitution, integration by analysis, andintegration by parts, or, as it is sometimes termed, successiveintegration.” [Google print search by James A. Landau]

INTERIOR ANGLE is found in English in 1629 in“A treatise of artificial fire-vvorks both for vvarres and recreation with divers pleasant geometricall obseruations,fortifications, and arithmeticall examples by Francis Malthus.[Google print search by James A. Landau]

INTERMEDIATE VALUE THEOREM.Theorem of intermediate value is found in English in 1902in the 10th ed. of theEncycopaedia Britannica in the article “Functions of Real Variables.” [Google print search by James A. Landau]

Intermediate value theorem appears in 1927 inThe Mathematics of Engineering byRalph Eugene Root. [Google print search]

The termINTERPOLATION was introduced into mathematics by JohnWallis (DSB; Kline, page 440).

The word appears in the English translation of Wallis' algebra(translated by Wallis and published in 1685), although the use thathas been found in the excerpt in Smith’sSource Book inMathematics appears not to be his earliest use of the term.

For some of the pre-history (and post-history) of interpolation see Erik MeijeringA Chronology ofInterpolation: From Ancient Astronomy to Modern Signal and Image ProcessingProceedings of the IEEE,90, 319-342.

INTERQUARTILE RANGE is found in 1882 in Francis Galton,"Report of the Anthropometric Committee,"Report of the 51stMeeting of the British Association for the Advancement of Science,1881, pp. 245-260: "This gave the upper and lower 'quartile'values, and consequently the 'interquartile' range (which is equal totwice the 'probable error') (OED2).

INTERSECTION (in set theory) is found inWebster’s NewInternational Dictionary of 1909.

INTO is used in the sense of a mapping in 1949 by Solomon Lefschetz,Introduction to Topology (OED).

INTRINSICALLY CONVERGENT SEQUENCE (Cauchy sequence) is apparently found in 1935 inAbstracts of Doctoral Dissertationsof the Ohio State University. [Google print search by James A. Landau]

The termINTRINSIC EQUATION was introduced in 1849 by WilliamWhewell (1704-1886) (Cajori 1919, page 324).

INTUITIONISM in the philosophyof mathematics. The term entered the English mathematical vocabulary with L.E. J. Brouwer’s inaugural lecture of 1912, which appeared in English as  "Intuitionismand Formalism,"Bull. Amer. Math. Soc.20, (1913), 81-96.  ThereBrouwer wrote

On what grounds the conviction of the unassailable exactnessof mathematical laws is based has for centuries been an object of philosophicalresearch, and two points of view may here be distinguished, intuitionism (largelyFrench) and formalism (largely German). ... The question where mathematical exactnessdoes exist, is answered differently by the two sides; the intuitionist says:in the human intellect, the formalist says: on paper. (pp. 82-3,OED)

SeeBrouwer in MacTutorandBrouwer in the Stanford Encyclopedia.See also the entriesFORMALISM andLOGICISM.

The termINVARIANT was coined by James Joseph Sylvester.It appears in 1851 in "On ARemarkable Discovery in the Theory of Canonical Forms and ofHyperdeterminants,"Philosophical Magazine, 4th Ser., 2,391-410: "The remaining coefficients are the two well-knownhyperdeterminants, or, as I propose henceforth to call them, the twoInvariants of the form ax⁴ + 4bx³y +6cx²y² + 4dxy³ + ey⁴."In the same article he wrote, "If I (a, b,..l) = I (a', b',..l'),then I is defined to be an invariant of f."

See alsonormal subgroup.

INVERSE (element producing identity element)appears in 1900 in James Pierpont "Galois' Theory of AlgebraicEquations. Part II. Irrational Resolvents,"Annals of  Mathematics. p. 48"For every elementT_κ exists an element(denote it byT_κ^-1) such thatT_κT_κ^-1 =T_κ^-1T_κ = 1.T_κ^-1 is calledthe inverse ofT_κ"  (OED2).

INVERSE (in logic) is found in 1890 inElements of Logic as a Science of Propositions byEmily Elizabeth Constance Jones. [Google print search by James A. Landau]

INVERSE of a matrix. In his fundamental work of 1858 "A Memoir on the Theory of Matrices"Arthur Cayley introducedtwo terms forL^-1, "or as it may be termed the inverse orreciprocal matrix."CollMath Papers, I, 480. Writers in English long preferred the termreciprocal matrix: see e.g. the famoustextbook by A. C. Aitken,Determinants and Matrices (1^st edition 1939, 9th edition 1956).However that term has disappeared andinverse matrix is now universal. This usage is more in line with that in abstractalgebra.

SeeMATRIX andGENERALIZED INVERSE.

INVERSE FUNCTION appears in in English in 1816 in thetranslation of Lacroix’sDifferential and Integral Calculus:"e^x and logx are inverse functions of eachother since log (e^x) =x" (OED2).

The termINVERSE GAUSSIAN DISTRIBUTION is found inM. C. K. Tweedie (1947) "Functions of a Statistical Variate with Given Means, with Special Reference to Laplacian Distributions,"Proceedings of the Cambridge Philosophical Society,43, 41-49. [Gerard Letac]

See alsoGAUSSIAN.

INVERSE PROBABILITY. From thetime of Augustus De Morgan’sEssay on Probabilities (1838) until thatof Harold Jeffreys’sTheory of Probability (1939) English works on probabilitydistinguished betweendirect andinverse probabilities. De Morgan(p. 53) distinguished between the two, "we have calculated the chances of anevent, knowing the circumstances under which it is to happen or fail. We arenow to place ourselves in an inverted position: we know the event, and ask whatis the probability which results from the event in favour of any set of circumstancesunder which the same might have happened." The alternative "sets of circumstances"were usually called "causes" or "hypotheses" and so the domain of inverse probabilitywas the probability of causes or what would now be called Bayesian probability.However the new Bayesian writers of the 1960s discarded the direct and inverseterms and these have fallen out of use.

A. I. Dale reports a much earlier use of the phrase "méthodeinverse des probabilités" in an outline Fourier wrote for some lectures givenat the École Polytechnique in Paris in the late 18^th or early 19^thcentury. (A History of Inverse Probability from Thomas Bayes to Karl Pearson.)

This entry was contributed by John Aldrich. SeeBAYES andPOSTERIOR PROBABILITY.

INVERSE VARIATION.Inverse ratio andinverselyare found in English in 1660 in Barrow’s translation of Euclid.

Inverse proportion is found in English in 1648 inThe construction and use of the logarithmeticall Tables byEdmund Wingate: “Probl. IV. Having three numbers given, to finde a fourth in an inverse proportion.”[Google print search by James A. Landau]

Inversely proportional is found in English in 1741 in a translation of Euclid by Dr. John Keil: “Coroll.From hence it is manifest, if four Magnitudes be proportional, that they will be also inversely proportional.”[Google print search by James A. Landau]

Varies inversely is found in 1789 inA System of Mechanics and Hydrostaticsby Rev. T. Parkinson: “Cor. 4.The latitude of a pulse [of sound] varies inversely as the number of vibrations of the sounding body in a given time, or directlyas the time of one vibration. . . .” [Google print search by James A. Landau]

Inverse variation is found in 1830 inThe Edinburgh Review,or Critical Journal: For April-July, 1830. “The law of inversevariation, or inverse proportion, is a much a part of mathematicalscience as the law of geometric progression. The only difference inthis respect between Mr Malthus and Mr Sadler is, that Mr Malthus knowswhat is meant by geometric progression, and that Mr Sadler has not thefaintest notion of what is meant by inverse variation. Had heunderstood the proposition which he has enounced with so muchpomp, its ludicrous absurdity must at once have flashed on his mind.”[Google print search by James A. Landau].

TheINVERTED GAMMA DISTRIBUTION is found in HowardRaiffa & Robert Schlaiffer’sApplied Statistical Decision Theory (1961, p. 227).

See alsoGAMMA DISTRIBUTION.

The termINVOLUTION is due to Gérard Desargues (1593-1662) (Kline, page 292).

IRRATIONAL. SeeRATIONAL AND IRRATIONAL.

IRREDUCIBLE INVARIANT is found in 1856 in “A Second Memoir upon Quantics” by Arthur Cayley in thePhilosophical Transactions.[James A. Landau]

ISOGON is a rare term for an equiangular polygon. TheOEDshows it in 1696 in the fifth edition of Edward Phillips' dictionary,where it is spelledisagon.

ISOGRAPHIC is the word used by Ernest Jean Philippe FauquedeJonquiéres (1820-1901) to describe the transformations he haddiscovered, later calledbirational transformations (DSB).

ISOMERIA is a term used by Vieta for freeing an equationof any fractions by multiplying both sides by the least commondenominator. The word is found in English in 1696 in Edward Phillips'dictionary.

ISOMETRY. Aristotle used the wordisometria.

Isometry is found in English inAppletons' Cyclopaedia of Drawing edited by W. E. Worthen, which is dated1857 but appears to be cited in a catalog printed in 1853 [University of Michigan Digital Library].

In its modern sense,isometry occurs in English in 1941 inSurvey of Modern Algebra by MacLane and Birkhoff: "An obviousexample is furnished by the symmetries of the cube. Geometricallyspeaking, these are the one-one transformations which preservedistances on the cube. They are known as 'isometries,' and are 48 innumber" (OED).

ISOMORPHIC, ISOMORPHISM. The termisomorphism was used early in crystallography. Some books on geology before 1864 referred to“geometrical isomorphism” or “mathematical isomorphism” between crystals.

Isomorphic was used by J. J. Sylvester in 1864 in“The Real And Imaginary Roots Of Algebraical Equations: A Trilogy,”in thePhilosophical Transactions:

To every point in space, it has been remarked, willcorrespond one particular family of equations all of the same characteras regards the number they contain of real or imaginary roots, becausecapable of being derived from one another by real linear substitutions,such family consisting of an infinite number of ordinary or conjugateequations according as the point is facultative or non-facultative; butit may be well to notice that, conversely, every point does notcorrespond to a distinct family. In fact every point in the curvesD =pJ²,L =qJ² (p, qbeing constants) will denote a curve divided into two branches by theorigin of coordinates, one of which will be facultative and the othernon-facultative; but in each separate branch every point will representthe very same family. Any such separate branch may be termed anisomorphic line; and we see that the whole of space may be conceived aspermeated by and made up of such lines radiating out from the origin inall directions.

In 1870Traité Des Substitutions Et Des Équations Algébriquesby Camille Jordan has (p. 56):

67. Un groupe Γ est ditisomorphe à un autre groupe G, si l'on peut établir entre leurs substitutions unecorrespondance telle : i° que chaque substitution de G corresponde à une seule substitution de Γ, et chaque substitution de Γ à une ou plusieurs substitutions de G; 2° que le produit de deux substitutions quelconques de G corresponde au produit de leurs correspondantes respectives. [Google print search]

A translation is: “A group Γ is called“isomorphic” to another group G,if one can establish between their substitutions a correspondence suchthat: i° each substitution of G corresponds to a singlesubstitution of Γ,and each substitution of Γ to one or more substitutions of G,2° the product of any two substitutions of G corresponds to theproduct of their respective corresponding substitutions.”

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

This entry was contributed by James A. Landau.

ISOSCELES. The isosceles triangle is treated inEuclid’sElementsXIII. xvi.The word is formed from equal-legged from +, leg.The word appears in English in Billingsley’stranslation ofEuclidI. Def. xxv. 5 “Isosceles, isa triangle, which hath onely two sides equall.”

A few years before, Recorde had introduced an English expression inThe Pathwaie to Knowledge: “There isalso 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 thyrdvnequall, which the Greekes callIsosceles, the Latine menaequicurio, and inenglish tweyleke [= two-like] may they be called.” See the entryEQUILATERAL for further discussion.

Billingsley adopted the Greek word but there was also a Latin word. TheOED reportsan isosceles triangle being calledanequicrure in 1644 (this usage it describes as obsolete) and anequicruraltriangle in 1650. These are the earliest uses it gives for the termsof Latin origin.

The termITERATED FUNCTION SYSTEM was coined by MichaelBarnsley, according to an Internet website.According to Wikipedia, IFSs were conceived in their present form by John E. Hutchinson in 1981and popularized by Michael Barnsley’s bookFractals Everywhere (1988).