*Last revision: April 30, 2016*

**ICOSAHEDRON.** The icosahedron is treated in
Euclid’s Elements XIII. xvi.
The word is formed from twenty + seat, base.
The English word is found in Sir Henry Billingsley’s 1570 translation of the *Elements* (*OED*).

**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 his edition of
P. G. L.
Dirichlet *Vorlesungen über Zahlentheorie* (ed. 2, 1871) Suppl. x. p. 452. According to Kline (p. 823), the name was
chosen “in honor of Kummer’s ideal numbers.” See also the article in the
*Encyclopaedia of Mathematics*.

The term passed without change into English. A *JSTOR* search produced references to
Dedekind’s theory of ideals in Arthur S. Hathaway “A Memoir in the Theory
of Numbers,” *American
Journal of Mathematics*, **9**,
(Jan., 1887), p. 166: “The theory of ideals here established, however, differs
from Dedekind’s theory in important respects…” [John Aldrich]

See the entry IDEAL NUMBER.

**IDEAL NUMBER.** According to Kline (p. 819), Ernst Eduard Kummer (1810-1893) created
a theory of ideal numbers in the papers he published in the
*Journal
für die reine und angewandte Mathematik,* **12**, 1847: “Zur
Theorie der complexen Zahlen” (pp. 319-326) and “Ueber die Zerlegung der aus
Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren” (pp.
327-367). See also the article in the *Encyclopaedia of Mathematics*.

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

**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 callednilpotent;but when, raised to a square or higher power, it gives itself as the result, it may be calledidempotent.

The defining equation of nilpotent and idempotent expressions are respectivelyA= 0, and^{n}A=^{n}A;but with reference to idempotent expressions, it will always be assumed that they are of the form

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

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
*E*^{2} = *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.]

The term **IDENTIFIABILITY** was coined by
the econometrician Tjalling C. Koopmans around 1945 with reference to the
economic *identity* of a relationship within a
system of relationships. (See SIMULTANEOUS EQUATIONS MODEL). The term began appearing in the econometrics literature
immediately, although Koopmans’s own exposition of the subject did not appear
until 1949--his "Identification Problems in Economic Model
Construction" *Econometrica*, **17**, pp. 125-144. Around 1950 the term was
taken up by statistical theorists and used in a more general sense, see e.g.
Jerzy Neyman’s "Existence of Consistent Estimates of the Directional
Parameter 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 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 s_{1} = identity,
s_{2}, 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*_{l}*e* such that, for every element
*a,* *i*_{l}*a* = *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).

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

**IFF.** The earliest citation for “iff” in the *OED* is from John L. Kelley’s *General Topology* (1955): “F is
equicontinuous at *x* iff there is a neighborhood of *x* whose image
under every member of F is small.” Kelley credited the term to
Paul R. Halmos.

On the last page of his autobiography, Halmos 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'tknowthat it did, and my invention (re-invention?) of it is what spread it through 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▐, 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”.

From *I Want to Be a
Mathematician: An Automathography,* by Paul R. Halmos, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985, page 403. See
Wolfram: iff.

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

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, xAn early appearance of the word^{3}- 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 x^{3}- 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.The quotation above is from Chapter LXVI (p. 264),

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 -a^{2} 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 1796
in *A Treatise of Algebra: In Three Parts*
by Colin Maclaurin. [Google print search]

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 found in English in 1836 in
*The differential and integral calculus*
by Augustus De Morgan.
[Google print search]

**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 INTEGRAL** occurs in "Concerning
Harnack’s Theory of Improper Definite Integrals" by Eliakim
Hastings Moore, *Trans. Amer. Math. Soc.,* July 1901.
The term may be much older.

**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).** The *Encyclopaedia Britannica* of 1771 has *contained angle*:
"Case IV. Two sides and the contained angle being given, to find the other two angles."

*Included angle* 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* appears in English in
1823 in *Dictionary 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** 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.** See DEPENDENT VARIABLE.

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

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).

**INDEX NUMBER** became a standard term in economic statistics in the 1880s. The OED’s earliest quotation is from 1875
William Stanley Jevons *Money and the Mechanism of Exchange*
chapter xxv p. 332:
"In the annual Commercial History and Review of the *Economist* newspaper, there has, for many years,
appeared a table containing the Total Index Number of prices, or the arithmetical
sum of the numbers expressing the ratios of the prices of many commodities to
the average prices of the same commodities in the years 1845-50." The table was the work of
William Newmarch but
Jevons had himself pioneered the method of index numbers in *A Serious Fall
in the Value of Gold Ascertained and its Social Effects Set Forth* (1863).
Anticipations of the technique have been found in the early 18^{th}
century, in the work of William
Fleetwood. See M. G. Kendall "The Early History of Index Numbers"
in M. G. Kendall & R. L. Plackett *Studies in the History of Statistics
and Probability*, volume 2 (1977).

**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 and BERNOULLI DISTRIBUTION.

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

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* Ì*X*) (*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.

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

According to the *DSB,* Christian Huygens used the term *infinitely 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’s *Logarithmotechnia,*
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.)](See Philip Beeley / Christoph J. Scriba (eds.),

Wallis used *infinitesima* 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 term *infinitesimal*
in 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’s
writings, 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 book *Logarithmotechnia* (1668) Mercator does not use the term
*infinitesimal,* but instead *pars 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, legat
infinitesimam.” This was a reaction to the critique by Wallis at
the beginning
of his review of the book, printed in the same issue of the *Philosophical Transactions,* pp. 753-759.

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

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

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

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

The term **INFINITESIMAL CALCULUS** is found in English in 1801 (MWCD11).

**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.** 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 pertinet ad priorem casum, et in omnibus speciebus curvæ Domini de Roberval (prima excepta) quæ pertinet ad 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 doctrina de maximis et minimis, hoc prwmisso lemmate generali:

The above was posthumously printed in the *Opera varia,* Toulouse, 1679, and is also in
Oeuvres
de Fermat, Paris : Gauthier-Villars, 1891, p. 166).

The inflection point of this curve had been mentioned by Roberval in his letter to Fermat from 22 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 est convexe en dedans, ce qui est admirable. Et encor plus que par le point B, il ne se peut mener de ligne droite qui la touche, mais une comme OBY, qui fera les angles au sommet OBA, YBC, chacun moindre qu'aucun angle rectiligne donne. Il en sera de mesme de l'autre part, et ce sont ces deux points d'où ie 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. 110-112
give more detailed information on the interchange between Huygens and
Schooten concerning this problem). The problem was later also solved by
Hendrik 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 duas
ejus portiones, concavam & convexam, à se invicem
distinguit."

The review of Sluse’s book in *Philosophical Transactions,* nr 45, (1669), p. 903-909, treats chapter 5
on 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 in *Philosophical Transactions,*
nr 90, (1673), p. 5143-5147,
Sluse refers to chapter 5 of the "Miscellanea": "qua ratione flexus
contrarii curvarum ex Tangentibus inveniantur, ostendi" (p. 5147).

In his 1684 article "Nova methodus pro maximis et minimis" in *Acta eruditorum,* Leibniz used
*punctum 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 page The History of Curvature.

*Point of Inflection* and *point of inflexion* are found in English in 1702 in *A 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.”

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.*"

This entry was largely contributed by Siegmund Probst, who found the material on Fermat and Roberval
in Cantor’s *Vorlesungen über Geschichte der Mathematik.*

**INFLUENCE CURVE** in robust statistical inference. The earliest *JSTOR*
appearance 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 later
Hampel’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 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. p. 709. 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, MEASURE OF** in the theory of communication was introduced by
Claude Shannon in his
*A Mathematical Theory of Communication*
(1948).

**INFORMATION THEORY.** The OED2 shows a number of citations for this term from 1950: e.g.
W. G. Tuller in *Trans. 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 of
C. E. Shannon,
in particular his *A Mathematical Theory of Communication* (1948).

See also BIT and ENTROPY.

**INJECTION, SURJECTION** and **BIJECTION.** The *OED* records a use of *injection*
by S. MacLane in the *Bulletin of the American Mathematical Society* (1950) and *injective* in Eilenberg
and Steenrod in *Foundations of Algebraic
Topology* (1952). However the family of terms is introduced on p. 80 of
Nicholas Bourbaki’s
*Théorie des ensembles,* Éléments de
mathématique Première Partie, Livre I, Chapitres I, II (1954). Reviewing the book in the
*Journal of Symbolic Logic*, R. O. Gandy (1959, p. 72) wrote:

Another useful function of Bourbaki’s treatise has been to standardise notation and terminology… 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 terms *did* prove acceptable, even to
mathematicians writing in English, and quickly became standard. For instance,
all three terms are used in Jun-Ichi Igusa “Fibre Systems of Jacobian
Varieties,” *American Journal of Mathematics*, **78**, (1956), 171-199.
(*JSTOR* search) The adjectival forms appear in C. Chevalley, *Fundamental
Concepts of Algebra* (1956): “A homomorphism which is injective is called a
monomorphism; a homomorphism which is surjective is called an epimorphism.” (*OED*)

See the entries ONE-TO-ONE and ONTO and MONOMORPHISM.

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 namesIn English,innerandouter.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).

**INSTRUMENTAL VARIABLES** in Econometrics and Statistics.
The term first appeared in Olav Reiersøl’s "Confluence Analysis by Means of Instrumental
Sets of Variables," *Arkiv for Mathematik, Astronomi och Fysik*, **32** (1945), 1-119.
In an ET interview (p.
119) Reiersøl recalled that the name was suggested by his
supervisor, Ragnar Frisch. Reiersøl
had proposed the method already in 1941 and independently R. C. Geary proposed
it in 1943. Reiersøl and Geary were interested in errors in variables models
but around 1950 it became clear that their method could be applied to the recently
developed SIMULTANEOUS EQUATIONS MODEL. Curiously the method had been used
in the 1920s by the economist Phillip G. Wright and his geneticist son Sewall
Wright for estimating demand functions but the method was not taken up and the
work had no influence in Econometrics. The relative contribution
of father and son has been a matter of controversy and Stock & Trebbi have
discussed the issue quite recently.

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

See the entries ERROR: ERRORS IN VARIABLES and PATH ANALYSIS.

**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.In the above quotation (th) represents a thorn. In this use,

*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).

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

The word **INTEGRAL** first appeared in print by Jacob 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 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 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 nameAccording to Smith (vol. 2, page 696), Leibniz in 1696 adopted the termcalculus summatoriusand the long letter [long S symbol] as the symbol. Bernoulli favored the namecalculus integralisand the capital letterIas 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 in 1911 in
*Monographs on Topics of Modern Mathematics: Relevant to the Elementary Field* by
numerous writers:
"Similarly, the integral domain [*R*_{1},
*R*_{2},
*R*_{3}, ...] may be defined by replacing
the expressions rational functions and rational coefficients in the
preceding definition by integral functions and integral coefficients respectively."
[Google print search]

**INTEGRAL EQUATION.** According to Kline (p. 1052) and J. Dieudonné *History of Functional Analysis* (p. 97), the term *integral
equation* (*Integralgleichung*)
is due to Paul du Bois-Reymond (1831-1889). Du Bois-Reymond introduced it in
his paper on the DIRICHLET
PROBLEM, “Bemerkungen über ,”
*Journal
für 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 were
collected into a book, *Grundzüge einer
allgemeinen Theorie der linearen Integralgleichungen* (1912).
Hilbert’s work played a critical part in the rise of FUNCTIONAL ANALYSIS.
See the *Encyclopaedia of Mathematics* entry Integral equations.

Although du Bois-Reymond introduced the term in its
modern sense, Euler had earlier used a phrase which is translated *integral equation* in the paper “De
integratione 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 integral
equations.” (*OED*).

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

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 1866 in
*Transactions of the Connecticut Academy of Arts and Sciences.*
[Google print search]

**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 1828 in *Elements of algebraical notation and expansion*
by George Walker:
"These two integrals being known, every integral of the form [an integral], where
*n* is a whole positive number, may be made to depend on one of these,
by a method of extensive use, called integration by parts, which we shall
now explain." [Google print search]

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 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 1927 in *The Mathematics of Engineering* by
Ralph Eugene Root. [Google print search]

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.

For some of the pre-history (and post-history) of interpolation see Erik Meijering
A Chronology of
Interpolation: From Ancient Astronomy to Modern Signal and Image Processing
*Proceedings of the IEEE*, **90**, 319-342.

**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.

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

**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).

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

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

See Brouwer in MacTutor and Brouwer in the Stanford Encyclopedia. See also the entries FORMALISM and LOGICISM.

The term **INVARIANT** was coined by James Joseph Sylvester.
It appears in 1851 in "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 ax^{4} + 4bx^{3}y +
6cx^{2}y^{2} + 4dxy^{3} + ey^{4}."
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 also *normal subgroup.*

**INVERSE (element producing identity element)**
appears in 1900 in James Pierpont "Galois' Theory of Algebraic
Equations. Part II. Irrational Resolvents," *Annals of Mathematics*. p. 48
"For every element *T*_{κ} exists an element
(denote it by *T*_{κ}^{-1}) such that *T*_{κ}*
T*_{κ}^{-1} = *T*_{κ}^{-1}
*T*_{κ} = 1. *T*_{κ}^{-1} is called
the inverse of *T*_{κ }" (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.[OED2]

**INVERSE** of a matrix. In his fundamental work of 1858 "A Memoir on the Theory of Matrices"
Arthur Cayley introduced *two* terms for *L*^{-1}, "or as it may be termed the inverse or
reciprocal matrix."
*Coll
Math Papers*, I, 480. Writers in English long preferred the term *reciprocal matrix*: see e.g. the famous
textbook by A. C. Aitken, *Determinants and Matrices* (1^{st} edition 1939, 9th edition 1956).
However that term has disappeared and *inverse matrix* is now universal. This usage is more in line with that in abstract
algebra.

See MATRIX and GENERALIZED INVERSE.

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

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

This entry was contributed by John Aldrich. See BAYES and POSTERIOR PROBABILITY.

**INVERSE FUNCTION** appears in in English in 1816 in the
translation of Lacroix’s *Differential and Integral Calculus*:
"*e ^{x}* and log

*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** is found in
M. 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 also GAUSSIAN.

**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 1824 in *An Excursion Through the United States
and Canada During the Years 1822-1823*
by William Newnham Blane: "To speak in mathematical language, the song of
birds varies inversely with their plumage." [Google print search]

*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.This citation was taken from the University of Michigan Digital Library [James A. Landau].

The **INVERTED GAMMA DISTRIBUTION** is found in Howard
Raiffa & Robert Schlaiffer’s *Applied Statistical Decision Theory*
(1961, p. 227).

See also GAMMA DISTRIBUTION.

**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.** See RATIONAL AND IRRATIONAL.

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

**ISOGON** is a rare term for an equiangular polygon. The *OED*
shows it in 1696 in the fifth edition of Edward Phillips' dictionary,
where it is spelled *isagon.*

**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).

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

**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).

**ISOMORPHIC, ISOMORPHISM.** The term *isomorphism* 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 the *Philosophical Transactions*:

To every point in space, it has been remarked, will correspond one particular family of equations all of the same character as regards the number they contain of real or imaginary roots, because capable of being derived from one another by real linear substitutions, such family consisting of an infinite number of ordinary or conjugate equations according as the point is facultative or non-facultative; but it may be well to notice that, conversely, every point does not correspond to a distinct family. In fact every point in the curvesD=pJ^{2},L=qJ^{2}(p, qbeing constants) will denote a curve divided into two branches by the origin of coordinates, one of which will be facultative and the other non-facultative; but in each separate branch every point will represent the very same family. Any such separate branch may be termed an isomorphic line; and we see that the whole of space may be conceived as permeated by and made up of such lines radiating out from the origin in all directions.

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

67. Un groupe Γ est ditisomorpheà un autre groupe G, si l'on peut établir entre leurs substitutions une correspondance 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 such that: i° each substitution of G corresponds to a single substitution 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 the product of their respective corresponding substitutions.”

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

This entry was contributed by James A. Landau.

**ISOSCELES.** The isosceles triangle is treated in
Euclid’s *Elements* XIII. xvi.
The word is formed from equal-legged from + , leg.
The word appears in English in Billingsley’s
translation of *Euclid* I. Def. xxv. 5 “Isosceles, is
a triangle, which hath onely two sides equall.”

A few years before, Recorde had introduced an English expression 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 [= two-like] may they be called.” See the entry EQUILATERAL for further discussion.

Billingsley adopted the Greek word but there was also a Latin word. The *OED* reports
an isosceles triangle being called
an *equicrure* in 1644 (this usage it describes as obsolete) and an *equicrural
triangle* in 1650. These are the earliest uses it gives for the terms
of Latin origin.

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