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

IDEAL (point or line) was introduced as idéal by J. V. Poncelet in Traité des Propriétés proj. des Figures (1822).

IDEAL (number theory) was introduced by Richard Dedekind (1831-1916) in P. G. L. Dirichlet Vorles. über Zahlentheorie (ed. 2, 1871) Suppl. x. 452 (OED2).

IDEAL NUMBER. Ernst Eduard Kummer (1810-1893) introduced the term ideale zahl in 1846 in Ber. über 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

unless it be otherwise distinctly stated.

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 (type of equation) is found in 1831 in the second edition of Elements of the Differential Calculus (1836) by John Radford Young: "This is obvious, for this first term is what the whole development reduces to when h = 0, but we must in this case have the identity f(x) = f(x); hence f(x) is the first term" [James A. Landau].

Young also uses the term identical equations in the same work.

IDENTITY (element) is found in 1894 in the Bulletin of the American Mathematical Society I: "Given an (abstract) group G_n ... with elements s₁ = identity, s₂, s_n (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 i_le such that, for every element a, i_la = 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 believe 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".

The earliest citation of "iff" in the OED2 is 1955 in General Topology by John L. Kelley:

F is equicontinuous at x iff there is a neighborhood of x whose image under every member of F is small.

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 réelles; mais quelquefois seulement imaginaires; c'est à dire qu'on peut bien toujiours en imaginer autant que aiy dit en chàsque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde à 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 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 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)

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.

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 suggestion appears in 1936 in his An Invitation to Mathematics.

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

The first edition of the Encyclopaedia Britannica (1768-1771) has: "Thus the square root of -a² cannot be assigned, and is what we call an impossible or imaginary quantity."

There are two modern meanings of the term imaginary number. In Merriam-Webster's Collegiate Dictionary, 10th ed., an imaginary number is a number of the form a + bi where b is not equal to 0. In Calculus and Analytic Geomtry (1992) by Stein and Barcellos, "a complex number that lies on the y axis is called imaginary."

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

The term IMAGINARY PART appears in 1836 in the second edition of Elements of the Differential Calculus by John Radford Young [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 théorie des définitions, Annales de Mathématique Pure et Appliquée (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 César de Araújo]

IMPLICIT DIFFERENTIATION is dated ca. 1889 in MWCD10.

IMPLICIT FUNCTION is found in 1814 New Mathematical and Philosophical Dictionary: "Having given the methods ... of obtaining the derived functions, of functions of one or more quantities, whether those functions be explicit or implicit, ... we will now show how this theory may be applied" (OED2).

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.

INCIRCLE was used in 1883 by W. H. H. Hudson in Nature XXVIII. 7: "I beg leave to suggest the following names: circumcircle, incircle, excircle, and midcircle" (OED2).

INCLUDED (angle or side) appears in 1806 in Hutton, Course Math.: "If two Triangles have Two Sides and the Included Angle in the one, equal to Two Sides and the Included Angle in the other, the Triangles will be Identical, or equal in all respects" (OED2).

In 1828, Elements of Geometry and Trigonometry (1832) by David Brewster (a translation of Legendre) has: "...Two triangles are equal when they have two angles and an interjacent side in each equal."

INCOMMENSURABLE. Incommensurability is found in Latin in the 1350s in the title De commensurabilitate sive incommensurabilitate motuum celi (the commensurability or incommensurability of celestial motions) by Nicole Oresme.

The term INDEFINITE INTEGRAL is defined by Sylvestre-François Lacroix (1765-1843) in Traité du calcul différentiel et integral (Cajori 1919, page 272).

Indefinite integral also appears in 1831 in Elements of the Integral Calculus (1839) by J. R. Young:

In the practical applications of the calculus, it is not the general, or, as it is usually called, the indefinite, integral that is ultimately required, because here the constant which completes the integral is indeterminate, whereas, in every particular inquiry this constant has a corresponding particular value, thus rendering the integral definite.

INDEPENDENT EVENT and DEPENDENT EVENT are found in 1738 in The Doctrine of Chances by De Moivre: "Two Events are independent, when they have no connexion one with the other, and that the happening of one neither forwards nor obstructs the happening of the other. Two events are dependent, when they are so connected together as that the Probability of either's happening is alter'd by the happening of the other."

INDEPENDENT VARIABLE is is found in the 1816 translation of Differential and Integral Calculus by Lacroix: "Treating the subordinate variables as implicit functions of the independent ones" (OED2).

INDETERMINATE FORM is found in An Elementary Treatise on Curves, Functions and Forces (1846) by Benjamin Peirce (1809-1880).

Forms such as 0/0 are called singular values and singular forms in in 1849 in An Introduction to the Differential and Integral Calculus, 2nd ed., by James Thomson.

In Primary Elements of Algebra for Common Schools and Academies (1866) by Joseph Ray, 0/0 is called "the symbol of indetermination."

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

INDICATOR. See totient.

INDICATOR FUNCTION and INDICATOR RANDOM VARIABLE. The term indicator of a set appears in M. Loève's Probability Theory (1955) and, according to W. Feller (An Introduction to Probability Theory and its Applications volume II), Loève was responsible for the term. Loève's Probability Theory did not use term indicator 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 Mathematical Society, 14, (1963), pp. 298-304. (See characteristic function of a set) [John Aldrich]

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. ]

INDUCTIVE (PARTIALLY) ORDERED SET. The adjective "inductive" used in this context was introduced by Bourbaki in Élements de mathématique. I. Théorie des ensembles. Fascicule de résultats, Actualités Scientifiques et Industrielles, no. 846, Hermann, Paris, 1939. Bourbaki's original term and definition is now standard among mathematicians: a poset (X, £) is inductive if every totally ordered subset of it has a supremum, that is:

(1) ("A Ì X) (A is a chain Þ A has a supremum).

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

(2) (" A Ì X) (A has a supremum),

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

(3) (" A ÌX) (A Æ and A has an upper bound Þ A has a supremum).

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

[Carlos César de Araújo]

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

A paper by Fermat is titled "La méthode de la 'descente infinie ou indéfinie.'" Fermat stated that he named the method.

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.

Point of inflexion appears in 1743 in Fluxions by Emerson: "The Point of Inflexion or contrary Flexure is that Point which separates the convex from the concave Part of the Curve" (OED2).

In An Elementary Treatise on Curves, Functions and Forces (1846), Benjamin Peirce writes, "When a curve is continuous at a point, but changes its direction so as to turn its curvature the opposite way at this point, the point is called a point of contrary flexure, or a point of inflexion."

INFORMATION, AMOUNT OF, QUANTITY OF in the theory of statistical estimation. R. A. Fisher first wrote about "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. 700-725. In the modern literature the qualification Fisher's information is common, distinguishing Fisher's measure from others originating in the theory of communication as well as in statistics. [John Aldrich and David (1995)].

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).

The term INNER PRODUCT was coined (in German as inneres produkt) by Hermann Günther Grassman (1809-1877) in Die lineale Ausdehnungslehre (1844).

According to the OED2 it is "so named because an inner product of two vectors 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 Günther 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.

The term INNUMERACY was popularized as the title of a recent book by John Allen Paulos. The word is found in 1959 in Rep. Cent. Advisory Council for Educ. (Eng.) (Ministry of Educ.): "If his numeracy has stopped short at the usual Fifth Form level, he is in danger of relapsing into innumeracy" (OED2).

INTEGER and WHOLE NUMBER. Writing in Latin, Fibonacci used numerus sanus.

According to Heinz Lueneburg, the term numero sano "was used extensively by Luca Pacioli in his Summa. Before Pacioli, it was already used by Piero della Francesca in his Trattato d'abaco. I also find it in the second edition of Pietro Cataneo's Le pratiche delle due prime matematiche of 1567. I haven't seen the first edition. Counting also Fibonacci's Latin numerus sanus, the word sano was used for at least 350 years to denote an integral (untouched, virginal) number. Besides the words sanus, sano, the words integer, intero, intiero were also used during that time."

The first citation for whole number in the OED2 is from about 1430 in Art 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 one of the earliest and most popular arithmetics in the English language, which appeared in 1537 at St. Albans. The work is anonymous, and its long title runs as follows: "An Introduction for to lerne to reken with the Pen and with the Counters, after the true cast of arismetyke or awgrym in hole numbers, and also in broken" (Julio González Cabillón).

Oresme used intégral.

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 for exactly nether by integer nor fraction it can be expressed" (OED2).

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

Whole number is most frequently defined as Z+, although it is sometimes defined as Z. In Elements of the Integral Calculus (1839) by J. R. Young, the author refers to "a whole number or 0" but later refers to "a positive whole 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 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 E. J. Finan, "A determination of the integral domains of the complete rational matric algebra of order 4," Bulletin A. M. S. (1930).

INTEGRAL EQUATION (calculus sense). According to Kline (page 1052) and Cajori 1919 (page 393), the term integral equation is due to Paul du Bois-Reymond (1831-1889), Jour. für 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].

Integral equation is found in English in 1802 in Woodhouse, Phil. Trans. XCII. 95: "Expressions deduced from the true integral equations" (OED2).

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

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

Integrand appears in the calculus sense in 1893 in A Treatise on the Theory of Functions by James Harkness and Frank Morley: "When c moves to t, the integrand of u₂ remains finite and continuous."

INTEGRATING FACTOR is found in May 1845 in a paper by Sir George Gabriel Stokes published in the Cambridge Mathematical Journal [University of Michigan Historical Math Collection].

INTEGRATION BY PARTS appears in 1831 in Elements of the Integral Calculus (1839) by J. R. Young: "...a formula which reduces the integration of udv to that of vdu, and which is known by the name of integration by parts."

The term appears in a paper by George Green under the heading "General Preliminary Results."

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

INTEGRATION BY SUBSTITUTION is found in about 1870 in Practical treatise on the differential and integral calculus, with some of its applications to mechanics and astronomy by William Guy Peck: "Integration by Substitution, and Rationalization. 67. An irrational differential may sometimes be made rational, by substituting for the variable some function of an auxiliary variable; when this can be done, the integration may be effected by the methods of Articles 65 and 66. When the differential cannot be rationalized in terms of an auxiliary variable, it may sometimes be reduced to one of the elementary forms, and then integrated" [University of Michigan Digital Library].

INTERIOR ANGLE is found in English in 1756 in Robert Simson's translation of Euclid: "The three interior angles of any triangle are equal to two right angles" (OED2).

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).

The word appears in the English translation of Wallis' algebra (translated by Wallis and published in 1685), although the use that has been found in the excerpt in Smith's Source Book in Mathematics appears not to be his earliest use of the term.

INTERQUARTILE RANGE is found in 1882 in Francis Galton, "Report of the Anthropometric Committee," Report of the 51st Meeting 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 to twice the 'probable error') (OED2).

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 INTRINSIC EQUATION was introduced in 1849 by William Whewell (1704-1886) (Cajori 1919, page 324).

INVARIANT appears in 1851 in James Joseph Sylvester, "On A Remarkable Discovery in the Theory of Canonical Forms and of Hyperdeterminants," Philosophical Magazine, 4th Ser., 2, 391-410: "The remaining coefficients are the two well-known hyperdeterminants, or, as I propose henceforth to call them, the two Invariants 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."

The term is due to Sylvester (1814-1897), according to Cajori (1919, page 345) and Kline (page 927), who supplies the reference Coll. Math. Papers, I, 273. Sylvester coined the term in 1851, according to Karen Hunger Parshall in "Toward a History of Nineteenth-Century Invariant Theory."

See also normal subgroup.

INVERSE (element producing identity element) appears in 1900 in Ann. Math. (OED2).

INVERSE (in logic) appears in 1896 in Welton, Manual of Logic:

Inversion is the inferring, from a given proposition, another proposition whose subject is the contradictory of the subject of the original proposition. The given proposition is called the Invertend, that which is inferred from it is termed the Inverse... The rule for Inversion is: Convert either the Obverted Converse or the Obverted Contrapositive.

INVERSE FUNCTION appears in in English in 1816 in the translation of Lacroix's Differential and Integral Calculus: "e^x and log x are inverse functions of each other since log (e^x) = x" (OED2).

Inverse function also appears in 1849 in An Introduction to the Differential and Integral Calculus, 2nd ed., by James Thomson: "A very convenient notation for expressing these and other inverse functions, as they have been called, has been proposed by Sir John Herschel."

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

INVERSE VARIATION. Inverse ratio and inversely are found in English in 1660 in Barrow's translation of Euclid.

Inverse proportion is found in 1793 in Beddoes, Math. Evid.: "A balance of which one arm should be ten inches, and the other one inch long, and each arm should be loaded in an inverse proportion to its length" (OED2).

Inversely proportional is found in Thomas Graham, "On the Law of the Diffusion of Gases," Philosophical Magazine (1833). The paper was read before the Royal Society in Edinburgh on Dec. 19, 1831: "Which volumes are not necessarily of equal magnitude, being, in the case of each gas, inversely proportional to the square root of the density of that gas." [James A. Landau]

Varies inversely is found in 1834 in M. Somerville, Connex. Phys. Sc. xxv. (1849): "The elasticity or tension of steam..varies inversely as its volume" (OED2).

Inverse variation is found in 1856 in Ray's higher arithmetic. The principles of arithmetic, analyzed and practically applied by Joseph Ray (1807-1855):

Variation is a general method of expressing proportion often used, and is either direct or inverse. Direct variation exists between two quantities when they increase togeether, or decrease together. Thus the distance a ship goes at a uniform rate, varies directly as the time it sails; which means that the ratio of any two distances is equal to the ratio of the corresponding times taken in the same order. Inverse variation exists between two quantities when one increases as the other decreases. Thus, the time in which a piece of work will be done, varies inversely as the number of men employed; which means that the ratio of any two times is equal to the ratio of the numbers of men employed for these times, taken in reverse order.

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 Gérard Desargues (1593-1662) (Kline, page 292).

IRRATIONAL. Cajori (1919, page 68) writes, "It is worthy of note that Cassiodorius was the first writer to use the terms 'rational' and 'irrational' in the sense now current in arithmetic and algebra."

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 Jonquiéres (1820-1901) to describe the transformations he had discovered, later called birational transformations (DSB).

ISOMETRIC. Isometrical is found in 1838 in the title Treatise on Isometrical Drawing by T. Sopwith (OED2).

Isometric is found in the Penny Cyclopaedia in 1840: "This specific application of projection was termed isometric by the late Professor Farish, who pointed out its practical utility, and the facility of its application to the delineation of engines, etc. ... A scale for determining the lengths of the axes of the isometric projection of a circle" (OED2).

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. Aristotle used the word isometria.

Isometry is found in English in Appletons' Cyclopaedia of Drawing edited by W. E. Worthen, which is dated 1857 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 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.