*Last revision: Nov. 24, 2015*

The **DANIELL INTEGRAL** was introduced by
Percy John Daniell
in “A General Form of Integral,” *Annals of Mathematics*, **19**,
(1918), 279-294. See the entry in
*Encyclopedia of Mathematics.*
The first person to use the Daniell Integral and to refer to it as such was
Norbert Wiener “The Mean of a Functional of Arbitrary Elements,” *Annals of Mathematics,* **22**, (1920), 66-72.

**DANIELL WINDOW** or **Daniell kernel** refers to a device used for smoothing the PERIODOGRAM
to obtain an estimate of the SPECTRAL DENSITY that was proposed by Percy John Daniell in a written contribution
to the discussion in the “Symposium on Autocorrelation in Time Series,” *Journal of the Royal Statistical Society, Supplement*, **8**, (1946), 88-90.

A Google print view shows a use of *Daniel’s window* which seems to be from 1958, but because only snippet view is available the date is uncertain.
*Gravity Measurements in Western Greenland,*
B. Lunos bogtr., Geodætisk institut (Denmark): “Cross correlation magnitude, cross power, coherency and phase have been computed with the aid of Daniell’s window
using different components of noise at the same station and also with like components from different stations.” [Google print search by James A. Landau]

A *JSTOR* search
found the phrase “the Daniell window” in M. B. Priestley “Evolutionary Spectra
and Non-Stationary Processes,” *Journal of the Royal
Statistical Society, B*, **27**, (1965), p. 220.

The term *Daniell kernel* is found in what appear to be documents from 1985 and 1987. [Google print search by James A. Landau]

See the entries WINDOW and KERNEL.

**DATA ANALYSIS**. This term had
been used casually for decades before John W. Tukey
made it his own in "The Future of Data Analysis," *Annals
of Mathematical Statistics*, **33**, (1962), 1-67. It is not exactly clear
what Tukey meant by *data analysis* and how it was different from *statistics*
but he quietly condemns a lot of statistical research. The paper concludes with
the proposition that the future of data analysis depends on "our willingness
to take up the rocky road of real problems in preference to the smooth road
of unreal assumptions, arbitrary criteria, and abstract results without real
attachments." (p. 64)

See STATISTICS and EXPLORATORY DATA ANALYSIS.

**DATA MINING**. Cheap computing has made it possible for statisticians and other researchers to investigate
alternative specifications and generally search for patterns in their data.
In the 40 or so years that the term *data mining* has been in use the scope
of the activity and its status have changed considerably. A *JSTOR* search
found this from 1962 "after a shameless amount of data-mining my best
candidate [equation] is ..." A. M. Okun "The Predictive Value of Surveys of Business Intentions,"
*American Economic Review*, **52**, Papers and Proceedings of the Seventy-Fourth
Annual Meeting of the AEA, p. 223. More recently D. J. Hand, G. Blunt, M. G.
Kelly & N. M. Adams write in their "Data Mining for Fun and Profit," *Statistical
Science*, **15**, (2000), 111-126, "Data mining is the process of seeking interesting or valuable information
in large data bases. ... [It] is an interdisciplinary subject, representing the
confluence of ideas from statistics, exploratory data analysis, machine learning,
pattern recognition, database technology, and other disciplines." (p. 111)

**DECAGON.** A Google print search shows what appear to be *latus decagon* and *decagon equilaterus* in a 1509 Latin translation of Euclid’s *Elements* by
Luca Pacioli. [Google print
search by James A. Landau]

*Decagon* appears in English in 1571 in *A geometrical
practise named Pantometria* by Thomas Digges (1546?-1595)
(although the spelling *decagonum* is used).

**DECILE** (in statistics) was introduced by Francis Galton
(Hald, p. 604).

*Decile* appears in 1882 in Francis Galton, *Rep. Brit. Assoc.
1881* 245: "The Upper Decile is that which is exceeded by
one-tenth of an infinitely large group, and which the remaining
nine-tenths fall short of. The Lower Decile is the converse of this"
(OED2).

A Google print search by James A. Landau finds pre-1882 uses of *decile* in astronomy and astrology.

**DECIMAL** is derived from the Latin *decimus,* meaning
"tenth." According to Smith (vol. 2, page 14), in the early printed
books numbers which are multiples of 10 were occasionally called
decimal numbers, citing Pellos (1492, fol. 4), who speaks of "numbre
simple," "nubre desenal," and "nubre plus que desenal" and Ortega
(1512, 1515 ed., fols. 4, 5), who has "lo numero simplice," "lo
numero decenale," and "lo numero composto."

In 1603 Johann Hartmann Beyer published *Logistica Decimalis.*

*Decimal* occurs in English in 1608 in the title *Disme: The
Art of Tenths, or Decimall Arithmetike.* This work is a
translation by Robert Norman of *La Thiende,* by Simon Stevin
(1548-1620), which was published in Flemish and in French in 1585.

**DECIMAL POINT.** In 1617 in his *Rabdologia* John Napier
referred to the *period* or *comma* which could be used to
separate the whole part from the fractional part, and he used both
symbols.

The 1689 edition of *Mr. Wingate’s Arithmetick* has: “A Decimal fraction may be express’d
without the Denominator, by prefixing a point or comma before (to wit, on the left hand of) the Numberator, so 5/10 may be written thus, .5 or thus, ,5, and 25/100 thus, ,25, or
thus, 25 [sic, should read “thus, .25”].” [Google print search by James A. Landau]

In 1704 the term *Separating Point* is used in the *Lexicon
Technicum* in the entry "Decimal."

In 1705 *Stereometry, Or, the ART of GAUGING,* 5th ed., by Thomas Everard has: “A *Decimal* Fraction, whether it stand
alone or be join’d with *whole* Numbers, is distinguished from a *whole* Number by a point or prick before it towards the left hand,
as in these Examples.” [Google print search by James A. Landau]

According to Cajori (vol. 1, page 329), "Probably as early as the time of Hutton the expression 'decimal point' had come to be the synonym for 'separatrix' and was used even when the symbol was not a point."

*Decimal point* appears in the 1771 edition of the
*Encyclopaedia Britannica* in the article "Arithmetick": "The
point thus prefixed is called the *decimal point*" [James A.
Landau].

In 1863, *The Normal: or, Methods of Teaching the Common Branches,
Orthoepy, Orthography, Grammar, Geography, Arithmetic and
Elocution* by Alfred Holbrook has: "The separatrix is the most
important character used in decimals, and no pains should be spared
to impress this on the minds of pupils."

**DECIMAL SYSTEM** is found in English in 1811 in *An Elementary
Investigation of the Theory of Numbers* by Peter Barlow [James A.
Landau].

**DECISION PROBLEM** (in mathematical logic) from the German *das Entscheidungsproblem.* According to S. C. Kleene *Introduction to
Metamathematics*, (1952, p. 136), “the *decision problem* … appears in modern logic with Schröder
1895, Löwenheim 1915 and Hilbert 1918.” The term *Entscheidungsproblem* is in the title of H. Behmann’s “Beiträge
zur Algebra der Logik, insbesondere zum Entscheidungsproblem,”
*Mathematische
Annalen* **86**, (1922), 163-239.

The German term appears in English texts,
not only in early works like Alonzo Church’s “A Note on the
Entscheidungsproblem,” *Journal of Symbolic
Logic*, **1**, (1936), 40-41
or A. M. Turing’s “On Computable Numbers with an Application to the *Entscheidungsproblem*,”
*Proceedings of the London Mathematical Society,* **32**, (1937), 232-265, but, more
surprisingly, in modern works. There has long been an acceptable English term,
“decision problem,”—see e.g. C. H. Langford *Journal
of Symbolic Logic*, **1**, (1936),
p. 111—but the two terms seem to co-exist quite happily.

See TURING MACHINE.

Statistical **DECISION THEORY** was essentially a 20^{th}
century development although some of the ideas can be found in earlier work:
see MEAN ERROR. The theory exists in both classical and Bayesian versions.

The classical theory was founded by Abraham Wald in 1939 ("Contributions
to the Theory of Statistical Estimation and Testing Hypotheses," *Annals
of Mathematical Statistics,* 10, 299-326) and developed in his book *Statistical
Decision Functions* (1950).

The phrase "decision theory" appears in E.
L. Lehmann's "Some Principles of the Theory of Testing Hypotheses,"
*Annals of Mathematical Statistics,* **21**, (1950), 1-26.

The first full presentation of the Bayesian
theory was Howard Raiffa & Robert Schlaiffer's *Applied Statistical Decision
Theory* (1961). The phrase "Bayesian decision theory" appeared in the titles
of two papers presented at the 1961 ASA conference see *American Statistician*,
**15**, (Oct., 1961), p. 25 [John Aldrich].

See also ADMISSIBILITY, BAYES ESTIMATE, BAYES RISK, LOSS FUNCTION, RISK FUNCTION, MINIMAX.

**DEEP (theorem).** From its
beginnings in Old English as a spatial term, the adjective *deep*
has acquired numerous figurative
meanings; see the *OED* for
details. In contemporary mathematics it is often applied to theorems:
“A theorem is deep whose proof is long, complicated, difficult, or
appears to involve branches of mathematics which are not obviously related to
the theorem itself.” D. Shanks *Solved and Unsolved
Problems in Number Theory*, 4th ed. (1993, p. 64).

The Hardy-Littlewood
school of analytic number theory were perhaps the first to use *deep* in the most esoteric of its
mathematical senses: for example, “It appears probable that such a result
cannot really be as deep as the prime number theorem, but nobody has succeeded
up to now in proving it by elementary reasoning.” writes S. Ramanujan “Highly
Composite Numbers,” *Proceedings of the London Mathematical Society* (1915), **14**,
(1), p. 386n. Here *elementary reasoning* means reasoning using
*only* concepts from the branch of
mathematics to which the theorem relates and, in particular, *not* using complex analysis to prove
results about prime numbers. See PRIME
NUMBER THEOREM.

The phrase *deep
theorem* seems to have come into print around 1930. “It is possible,
at heavy cost, to improve slightly upon the theorem [...] but I am not aware that
a proof has been given without employing [...] a deep theorem due to Walfisz.”
writes J. R. Wilton “A Series of Bessel Functions Connected with the Lattice-Points
of an *n*-Dimensional Ellipsoid”, *Proceedings of
the Royal Society A*, **120**, (Sept.
1928), p. 361.

[This entry was contributed by John Aldrich.]

The term **DEFINITE INTEGRAL** was apparently introduced by Laplace. It appears in his
"Memoire sur les suites"
pp. 207-309 *Histoire de l'Académie Royale des Sciences*. Paris, 1782 (year 1779). p. 209: "je
nomme *intégrale définie*, une intégrale prise depuis une valeur déterminée
de la variable jusqu'à une autre valeur déterminée." [Udai Venedem]

The term *definite integral* is defined by Sylvestre-François Lacroix (1765-1843) in
*Traité du calcul différentiel et integral,
Tome second* (1798) (Cajori 1919, page 272.)

Augustin Louis Cauchy (1789-1857) used the term. It is found in his
"Intégrales définies,"
*Oeuvres* (2), IV, 125: "This limit is called a definite integral."

*Definite integral* is found in English in 1831 in *Elements
of the Integral Calculus* (1839) by J. R. Young:

Thus, if we know that forInx=athe value of the integral Fx+ C ought to be A, then we have Fa+ C = A, therefore the value of the constant is in that case C = A - Fa,so that thedefinite integral,as it is then called, is Fx+ A - Fa.

In 1922 *Introduction to the Calculus* by William F. Osgood has "Thus the definite
integral is, *by definition,* the limit of a sum...."

Although modern textbooks generally define *definite integral* as the limit of a sum,
*Webster's Third New International Dictionary* (1961), defines it
as
"the number obtained by finding an antiderivative of a function,
substituting for the variable the first of two given numbers, then
the second, and subtracting the second result from the first."

**DEGREE (angle measure)** is found in English in about 1386 in
Chaucer's *Canterbury Tales:* "The yonge sonne That in the Ram
is foure degrees vp ronne" (OED2). He again used the word in about
1391 in *A Treatise on the Astrolabe:* "9. Next this folewith
the cercle of the daies, that ben figured in manere of degres, that
contenen in nombre 365, dividid also with longe strikes fro 5 to 5,
and the nombre in augrym writen under that cercle."

**DEGREE (of a polynomial).** See *order.*

**DEGREES OF FREEDOM**. The term *degrees of freedom* is found in a non-statistical sense
in, for example, 1867 in *Treatise
on Natural Philosophy*
by Sir William Thomson and Peter Guthrie Tait:
“Several references have been made in preceding sections to the number of independent variables in a displacement, or to the degrees of *freedom* or *constraint* under
which the displacement takes place. ... A free point has *three* degrees of freedom, inasmuch as the most general displacement which it can take is resolvable into three,
parallel respectively to any three directions, and independent of each other.” [Google print search by James A. Landau]

Fisher introduced degrees of freedom in connection with Pearson’s χ^{2} test
in the 1922 paper
“
On the Interpretation of *χ ^{2}* from
Contingency Tables, and the Calculation of P

See also chi-squared, *F*-distribution
and Student's *t*-distribution.

**DEL** (as a name for the symbol )
is found in 1901 in *Vector Analysis, A text-book for the use of students of
mathematics and physics founded upon the lectures of J. Willard Gibbs* by Edwin
Bidwell Wilson (1879-1964):

There seems, however, to be no universally recognized name for it, although owing to the frequent occurrence of the symbol some name is a practical necessity. It has been found by experience that the monosyllableAccording to Stein and Barcellos (page 836), this is the first appearance in print of the worddelis so short and easy to pronounce that even in complicated formulae in which (the symbol) occurs a number of times no inconvenience to the speaker or hearer arises from the repetition (OED2).

See NABLA (another name for the same operator) and the Earliest Uses of Symbols of Calculus page.

The term **DELTAHEDRON** is found in 1871 in *A Handbook to the Mineralogy of Cornwall and Devon*
by J. H. Collins, which has an illustration of a “deltahedron” with the note that it is the crystal form of Garnet, Argentite, and Analcite. [Google print search by James A. Landau]

In 1873 *A First Book of Mineralogy* by J. H. Collins
has several occurences of *deltohedron,* which is defined as a polyhedron bounded by twenty-four equal deltoids, where a *deltoid*
is defined on page 17 as “a four-sided figure of such a form that one of its diameters will divide it into two *equal scalene triangles,* and the other into two *unequal isosceles*
triangles.” [Google print search by James A. Landau]

**DEMOIVRE'S THEOREM.** In his *Synopsis Palmariorum
Matheseos* (1707), W. Jones refers to a "Theorem" of "that
Ingenious Mathematician, *Mr. De Moivre.*"

*Theorema Moivræanum* appears in a review of Jones's book
in *Acta Eruditorum* (1707).

However, the above use of the term refers to a different theorem from
the one now associated with this term, and according to Smith in *A
Source Book in Mathematics,* "The terms De Moivre's Formula, De
Moivre's Theorem, applied to the formula we are considering, do not
seem to have come into general use till the early part of the
nineteenth century." The remaining citations pertain to what is now
known as Demoivre's formula.

*Demoivre's formula* appears in A. L. Crelle, *Lebrbuch der
Elemente der Geometrie und der ebenen und spbärischen
Trigonometrie,* Berlin, vol. I, 1826, according to Tropfke.

The earliest use in English of *Demoivre’s formula* which unmistakably refers to
(cis θ)^{n} = cis *n*θ which James A. Landau could find in a Google print search is in 1809 in
*Treatise on Plane and Spherical Trigonometry* by Robert Woodhouse.

The earliest use in English of *Demoivre’s theorem* which unmistakably refers to
(cis θ)^{n} = cis *n*θ which James A. Landau could find in a Google print search is in 1812 in
*A Treatise on Trigonometry and on Trigonometrical Tables and Logarithms* by J. Hymers.

**DE MORGAN’S LAWS.** According
to W. & M. Kneale *The Development of Logic* (1962) p. 295 these were
known to medieval logicians and "occur explictly" in
William of Ockham's
(c.1287-1347) *Summa Totius Logicae*. They were rediscovered by
Augustus de Morgan
(1806-1871) and appear in "On the Symbols of Logic, the Theory of the Syllogism, and in particular
of the Copula," *Transactions of the Cambridge Philosophical Society,*
**9**, (1850): "The contrary of an aggregate is the compound of the contraries
of the aggregants: the contrary of a compound is the aggregate of the contraries
of the components." (p. 119 *On the Syllogism and Other Logical Writings
by Augustus De Morgan* edited by Peter Heath.)

The mathematical logicians Peano and Schröder referred to De
Morgan. The progress of De Morgan terms in English can be followed through the
*OED* and *JSTOR*. The *OED* quotes C. I. Lewis *Survey of
Symbolic Logic* (1918, p. 125): "3.4 and 3.41 together state De Morgan's
Theorem." The first *JSTOR* appearance is in 1945 when "De Morgan’s Laws"
is an entry in an index in the *Journal of Symbolic Logic*.

**DENSE** is found in English in 1902 in *Proc. Lond. Math.
Soc.* XXXIV: "Every example of such a set [of points] is
theoretically obtainable in this way. For..it cannot be closed, as it
would then be perfect and nowhere dense."

**DEPENDENT/INDEPENDENT VARIABLE.** *Independent variable* appears in 1813 in *Memoirs of the Analytical Society*
(Cambridge): “The method of Laplace for reducing an equation of the first order, where the difference of the independent variable is any function of the variable itself, to one
wherein that difference is constant, is well known.”
[Google print search by James A. Landau]

*Subordinate variable* and *independent variable* appear in English in the 1816 translation of *Differential
and Integral Calculus* by Lacroix: "Treating the subordinate variables
as implicit functions of the indepdndent [sic] ones" (OED2).

*Dependent variable* appears
in in 1831 in the second edition of *Elements of the Differential Calculus*
(1836) by John Radford Young: "On account of this dependence of the value
of the function upon that of the variable the former, that is *y,* is called
the *dependent* variable, and the latter, *x,* the *independent*
variable" [James A. Landau].

In Statistics R. A Fisher used the terms *dependent* and *independent*
in his presentation of regression analysis: see Section 25 of
Statistical Methods for Research Workers
(1925) on "Regression coefficients". For Fisher these terms qualified
*variate* but later writers have generally favoured *variable*.

See also REGRESSION.

**DERANGEMENT.** An early use of this term is in W. A. Whitworth, *Choice and Chance,* Cambridge, 1870 where it means a nonidentity permutation. P. A. P. MacMahon uses it in its modern sense of a fixed-point-free permutation in “The problem of derangement in the theory of permutations,” *Trans. Camb. Phil. Soc.,* Vol 21, pp. 467-481, 1908.
[David Callan]

**DERIVATIVE.** See the entry DIFFERENTIAL CALCULUS, DERIVATIVE.

**DESCARTES’ RULE OF SIGNS.** In 1809 *The Gentleman’s Mathematical Companion for the year 1798,
The Second Edition* has “Descartes’ rule.” [Google print search by James A. Landau]

The 1824 *Supplement to the Fourth, Fifth, and Sixth Editions of the Encyclopædia Britannica,*
Volume Fourth, has “rule of Descartes.” [Google print search by James A. Landau]

In 1833
*The penny cyclopædia* has:
“The latter of the two
works contains an extension of Descartes’ well known rule of signs,
by means of which the number of the real roots of an equation may be determined.”
[Google print search]

*London and Edinburgh Philosophical Magazine and Journal of Science,*
Vol. VII,
July-December 1835 has “the *rule of signs* of Descartes.” [Google print search by James A. Landau]

*Descartes' rule of signs* is found in English in 1855 in
*An elementary treatise on mechanics, embracing the theory of statics and dynamics,
and its application to solids and fluids. Prepared for the undergraduate course
in the Wesleyan university*
by Augustus W. Smith:
"Now, since the degree of the equation is even and the absolute term is negative,
there are at least two possible roots, one positive and the other negative.
The other two roots may be real or imaginary. If real, Descartes' rule of
signs indicates that three will be positive and one negative"
[University of Michigan Digital Library].

The term **DESCRIPTIVE GEOMETRY** occurs in the title
*Géométrie Descriptive* (1795) by Gaspard Monge
(1746-1818).

**DESCRIPTIVE STATISTICS.** The term entered circulation in the 1920s under the influence of a
discussion in J. M. Keynes's *A Treatise on Probability* (1921, p. 327).

The first function of [the Theory of Statistics] is purelydescriptive. It devises numerical and diagrammatic methods by which certain salient characteristics of large groups of phenomena can be briefly described; […] The second function of the theory isinductive. It seeks to extend its description of certain characteristics of observed events to the corresponding characteristics of other events which have not been observed.

The distinction became established although later writers usually preferred *inferential* to *inductive*.
Keynes and many of these writers believed that
the descriptive function had been over-emphasised.

(The term *descriptive statistics* is found in use as early as 1835, but not in the modern sense
in which it is paired with *inferential statistics.*)

Compare DATA ANALYSIS and EXPLORATORY DATA ANALYSIS.

**DESIGN MATRIX** in the analysis of variance and regression. Around 1950 several
authors began using matrix formalism in describing the design of experiments.
The problem of design was interpreted as the choice of a "design matrix" as in
K. D. Tocher’s "The Design and Analysis of Block Experiments," *Journal of the Royal
Statistical Society B*, **14**, (1952), p. 48.
More recently the term has been used for the *X*
matrix (in the usual formalism) regardless of whether an experiment is involved. E.g. the term has been used in
this way by econometricians since the 1970s: see e.g. G. Judge & M. E. Bock
"A Comparison of Traditional and Stein-Rule Estimators under Weighted Squared
Error Loss." *International Economic Review*, **17**, (1976), p. 238. (*JSTOR* search)

This entry was contributed by John Aldrich. See the entry ERROR in the usual formalism.

**DESIGN OF EXPERIMENTS** in Statistics. The phrase and the subject it refers to are associated with the 1935
book by R. A. Fisher, *The Design of Experiments*, although Fisher had already been working on the subject
for more than a decade. His earliest published thoughts are in chapter VIII,
section 48, "Technique of Plot Experimentation," of
*Statistical
Methods for Research Workers* (1925). He gave a fuller account in
"The
Arrangement of Field Experiments", *Journal of the Ministry of Agriculture of Great Britain,* **33**, (1926)

Other entries related to the design of experiments: ALIAS, BLOCK, CONFOUNDING, COVARIANCE, COVARIATE, DESIGN MATRIX, EFFECTS, ERROR, FACTOR, LATIN SQUARE, TREATMENT.

The term **DESMIC** was coined by Cyparissos Stephanos (1857-1918)
in "Sur les systemes desmiques de trois tetraedres," published in
*Darboux's Bulletin* ser 2, vol 3 (1879), pp 424-456 [Julio
González Cabillón, Michael Lambrou].

**DETERMINANT (discriminant of a quantic)** was introduced in 1801 by Carl Friedrich Gauss in his
*Disquisitiones arithmeticae* *Werke*
Bd. 1, p. 122:

Numerumbb—ac,a cuius indole proprietates formae (a, b, c) imprimis pendere in sequentibus docebimus,determinantemhuius formae uocabimus.

(Cajori vol. 2, page 88; Smith vol. 2, page 476).

Laplace had used the term *resultant* in this sense (Smith, 1906).

**DETERMINANT (modern sense).** Thomas Muir begins his monumental *The
Theory of Determinants in the Historical Order of Development* with
Leibniz in 1693 and describes about 20 contributions before he reaches 1812
when Augustin-Louis Cauchy (1789-1857) first used *determinant*
in its modern sense (Schwartzman, page 70). Cauchy
employed the word in “Memoire sur les fonctions qui ne peuvent obtenir que
deux valeurs egales et des signes contraires par suite des transpositions
operees entre les variables qu'elles renferment”, addressed on November
30, 1812, and first published in *Journal de
l'Ecole Poytechnique,* XVIIe Cahier, Tome X, Paris, 1815
Oeuvres (2) i: p. 112. Cauchy
refers to Gauss’s use of the term (see preceding entry)

M. Gauss s'en est servi avec avantage dans sesRecherches analytiquespour decouvrir les proprietes generales des formes du second degre, c'est a dire des polynomes du second degre a deux ou plusieurs variables, et il a designe ces memes fonctions sous le nom dedeterminants.Je conserverai cette denomination qui fournit un moyen facile d'enoncer les resultats; j'observerai seulement qu'on donne aussi quelquefois aux fonctions dont il s'agit le nom de resultantes a deux ou a plusieurs lettres. Ainsi le deux expressions suivantes,determinantetresultante,devront etre regardees comme synonymes.

(Smith vol. 2, page 477; Julio González Cabillón.)

Binet used the term RESULTANT in a memoir presented on the same occasion.

For the word in English the earliest citation in the *OED* is Cayley
“On the Theory of Determinants,” 1843 reprinted in
*Papers
vol. 1, p.63*.

**A list of matrix and linear algebra terms having entries on this web site is
here.**

**DEVIANCE.** The term was introduced by J. A. Nelder & R. W. M. Wedderburn in their fundamental paper on
GENERALIZED LINEAR MODELS, viz. "Generalized Linear Models,"
*Journal of the Royal Statistical Society, A*, **135**, (1972), 370-384. They write,
"The fitting of the parameters at each stage is done by maximizing the likelihood
for the current model and the matching of the model will be measured
quantitatively by the quantity -2*L*_{max} which we propose to call the *deviance*."
(p. 375) (David (1998))

**DIAGONAL.** Julio González Cabillón says, "Heron
of Alexandria is probably the first geometer to define the term
*diagonal* (as the straight line drawn from angle to angle)."

**DIALYTIC** was used by James Joseph Sylvester (1814-1897)
in 1853 in *Phil. Trans.* CXLIII. i. 544:

Dialytic. If there be a system of functions containing in each term different combinations of the powers of the variables in number equal to the number of the functions, a resultant may be formed from these functions, by, as it were, dissolving the relations which connect together the different combinations of the powers of the variables, and treating them as simple independent quantities linearly involved in the functions. The resultant so formed is called the Dialytic Resultant of the functions supposed; and any method by which the elimination between two or more equations can be made to depend on the formation of such a resultant is called a dialytic method of elimination.

**DIAMETER.** According to Smith (vol. 2, page 278), “Euclid used the word 'diameter' in relation
to the line bisecting a circle and also to mean the diagonal of a square, the
latter term being also found in the works of Heron." The first reference
is to Euclid Book 1, definition
17 and the second to proposition
34.

Euclid’s word *diametros* (according to Schwartzman composed
of “across” and “measure”) was adopted into Latin. The
English word appears in 1387 in John de Trevisa’s translation of Bartholomaeus Anglicus
*De Proprietatibus Rerum*, “þe dyameter [of] a
figure [is] þe lengest even lyne þat is devysed þerynne, take who þat may.” Robert Recorde’s
*Pathway to Knowledge* of 1551 has “Def., And all the lines that bee drawen crosse the circle, and goe by
the centre, are named diameters.” (*OED*).

See RADIUS.

**DIFFERENTIAL CALCULUS, DERIVATIVE.** The differential calculus required
new terms when it was invented more than 300 years ago and it has gone on
inspiring new terms ever since. Of course not all the terms that were coined remain in use, e.g.
Newton’s term **fluxion** has not survived; see
the entry FLUENT and FLUXION.
See also Earliest Use of Symbols of Calculus.

Many of the basic terms were introduced by Gottfried
Wilhelm Leibniz (1646-1716) writing in Latin. Because
of the strength of the Newtonian tradition in England many of these terms
entered English only in the 18^{th} or 19^{th} centuries.

**DIFFERENTIAL CALCULUS.** The term *calculus differentialis* was introduced by Leibniz in 1684 in
*Acta Eruditorum 3.* Before introducing this term, he used the expression *methodus tangentium directa* (Struik, page
271). The *OED* has a nice
quotation from Joseph Raphson’s *Mathematical
Dictionary* of 1702: “A different way....passes....in
France under the Name of Leibnitz's Differential Calculus, or Calculus of
Differences.” See also the entry CALCULUS.

**DIFFERENTIAL EQUATION.** Leibniz used the Latin *aequationes
differentiales* in *Acta
Eruditorum,* October 1684. See the entry ALGORITHM for the context.

*Differential equation* is found in English in 1749 in James Stirling, *The Differential Method, or a Treatise Concerning Summation and Interpolation of Infinite Series, Translated into English, with the Author's Approbation, by Francis Holliday.*
[Google print search, James A. Landau]

**DIFFERENTIAL (noun)** appears in the title of a manuscript of Sept. 10, 1690, by Leibniz,
“Methodus pro differentialibus, ponendo *z* = *dy:dx* et quaerendo *dz* ["A method for differentials,
positing *z* = *dy/dx* and seeking *dz*”]. He may have used the term
earlier, since he used the terms “differential equation” and
“differential calculus” earlier.

*Differential* appears in English as a noun in 1704 in *Lexicon technicum, or an universal English dictionary of arts and
sciences* by John Harris (OED2).

For the modern interpretation see the entry FRÉCHET DIFFERENTIAL.

Leibniz’s immediate successors, the Bernoullis and
Euler, also wrote in Latin but by the middle of the 18^{th} century
French was becoming a major mathematical language. In modern English, when we
speak of obtaining the **derivative** of a function by the process of **differentiation**,
we are combining Lagrange’s French and Leibniz’s Latin.

**DERIVATIVE.** Joseph
Louis Lagrange (1736-1813) used *derivée
de la fonction* and *fonction
derivée de la fonction* as early as 1772 in “Sur une nouvelle
espece de calcul relatif a la différentiation et a l'integration des quantités
variables,” *Nouveaux Memoires de
l'Academie royale des Sciences etBelles-Lettres de Berlin.*
(Oeuvres, Vol. III)
pp. 441-478. Lagrange states, for instance (first pages):

...on designe de même par u'' une fonction derivée de u' de la même maniére que u' l'est de u, et par u''' une fonction derivée de même de u'' et ainsi, ...... les fonctions u, u', u'', u''', u

^{IV}, ... derivent l'une de l'autre par une même loi de sorte qu'on pourra les trouver aisement par une meme operation répetée. [the functions u, u', u'', u''', u^{IV}, ... are derived one another from the same law, such that ...]

Lagrange used these terms in his
*Théorie
des fonctions analytiques* (1797). He introduced
the cognate term *primitive*: see the entry PRIMITIVE.

Not all French authors followed Lagrange.
Sylvestre-François Lacroix
(1765-1843) used the term *coefficien différential* in his
*Traité
du calcul différentiel et integral* p. 98 (Cajori 1919, page 272).

Lacroix’s book had a significant place in the history of mathematics in England
for it was the first Continental calculus book to be translated. In
1816 it was translated by members of the Analytical Society,
Babbage,
Herschel and
Peacock.
**DIFFERENTIAL COEFFICIENT** appears in the translation: the *OED* quotes the passage:
“The limit of the ratio of the increments, or the
differential coefficient, will be obtained” (OED2).The word **DIFFERENTIATE**
also appears: “The differential coefficient
being a new function....may itself be differentiated.” (*OED*).

The term **DERIVATIVE** (from Lagrange)
made occasional appearances in English texts of the 19^{th} century: in 1834 W. R. Hamilton referred to the
“derivative or differential coefficient” in his
On a General Method in Dynamics
*Philosophical Transactions of the Royal Society*. But it was not the favoured term.
In his *Differential and Integral Calculus* (1891) George A. Osborne used the term *differential
coefficient* but acknowledged, “The differential coefficient is
sometimes called *the derivative.*”
In the 20^{th} century the preference has been reversed: see G. H. Hardy
*A Course of Pure Mathematics* (1908, p. 197).

In the 19^{th} and 20^{th} centuries new subjects with new constructions appeared.
Besides those below see ABSOLUTE
DIFFERENTIAL CALCULUS or TENSOR ANALYSIS.

According to Kline (p. 554), the term **DIFFERENTIAL GEOMETRY** was first used by
Luigi
Bianchi (1856-1928) in 1894; his 1894 book
*Lezioni
di geometria differenziale* had existed in a preliminary form
since 1886. A major influence on Bianchi’s conception of the subject was
Gauss’s work on surfaces (ca. 1825): see the entry GAUSSIAN CURVATURE.
Kline’s account of the history of differential geometry begins in
the 17th century with Huygens.

According to an announcement in the *American Mathematical Monthly*, **6**,
(1899), p. 158, Clark University was offering a course in differential geometry
in 1899-1900. (*JSTOR*)

**DIFFERENTIAL FORM.** Victor J. Katz writes, “The mathematician
chiefly responsible for clarifying the idea of a differential form was
Elie Cartan.
In his fundamental paper “Sur certaines
expressions différentielles et le problème de Pfaff,”
*Annales
Scientifiques de l'École Normale Supérieure* (3)
tome 16 (1899) 239-332, Cartan defines an “expression
différentielle” as a symbolic expression given by a finite number of sums and
products of the *n* differentials,
and certain coefficient functions of the variables
*x*_{1,}
*x*_{2,}
*, …, *
*x*_{n}.”
Katz adds that Cartan first used the terms “exterior differential form”
and “exterior derivative” in his Leçons sur les invariants intégraux 1922.
(Katz “The History of Stokes’ Theorem,” *Mathematics Magazine*, **52**, (1979), 146-156.)

The term **DIFFERENTIAL TOPOLOGY** appears in
Marston
Morse’s “Closed Extremals,” *Annals of Mathematics*, **32**,
(1931), pp. 549-566 but, to judge from *JSTOR*, the term was not widely used before the 1960s. Differential topology first
appears as a major category in *Mathematical Reviews* (one of 95) in 1961.

See TOPOLOGY.

**DIFFERENTIABLE MANIFOLD.** The term was introduced by
Hassler
Whitney "Differentiable Manifolds," *Annals of Mathematics*, **37**, (1936), 645-680. Whitney remarks that
"differentiable manifolds have been studied, for instance, by Oswald Veblen and J. H. C. Whitehead
*The Foundations of Differential Geometry* ... 1932." However, neither the term nor the precise construction
appears there.

**DIFFEOMORPHISM** appears in
John
Milnor “On Manifolds Homeomorphic to the 7-Sphere,”
*Annals of Mathematics*, **64**, (1956), pp. 399-405 with the
explanation “A *diffeomorphism f* is a homeomorphism onto, such
that both *f* and *f* ^{-1}are differentiable.”
(p. 401, fn 3.) (*JSTOR* search)

See HOMEOMORPHISM.

For a list of entries associated with calculus see here.

**DIGIT.** According to Smith (vol. 2, page 12), the late Roman
writers seem to have divided the numbers below 100 into *digiti*
(fingers), *articuli* (joints), and *compositi* (composites
of fingers and joints).

In English, Robert Recorde in the 1558 edition of the *Ground of Artes* wrote, "A diget is any numbre vnder 10."

The term **DIGITADDITION** was coined by D. R. Kaprekar, according to an Internet web page.

**DIGRAPH** was used in 1955 by F. Harary in *Transactions of
the American Mathematical Society.* The term *directed graph*
also occurs there (OED2).

**DIHEDRAL** appears in 1799 in George Smith, *The laboratory:
or school of arts*: "Terminating in dihedral pyramids" (OED2).

**DIHEDRAL ANGLE** appears in 1826 in Henry, *Elem. Chem.*:
"Variations of temperature produce a ... difference in ... a crystal
of carbonate of lime.... As the temperature increases, the obtuse
dihedral angles diminish ...so that its form approaches that of a
cube" (OED2).

**DIOPHANTINE ALGEBRA** is found in English in 1752 in *The Elements of Algebra in a New and Easy method; Second Edition*
By Nathaniel Hammond: “In this Piece [referring to a German Algebra treatise] there are a great many curious Things relating especially to *Diophantine Algebra*.”
[Google print search by James A. Landau].

**DIOPHANTINE ANALYSIS** (named for
Diophantus
of Alexandria)
occurs in French in a letter of March
1770 from Euler to Lagrange: “ce problème me paraissait d'une nature singulière
et surpassait même les règles connues de l'analyse de Diophante” (“this problem
appeared to me to be of a singular nature and surpassed the known rules of
Diophantine analysis”).

Lagrange used “analyse de Diophante” in a letter to D’Alembert in June 1771.

*Diophantine analysis* is found in English 1795 in *A Mathematical and Philosophical Dictionary* [Google print search].

[This entry was contributed by James A. Landau.]

**DIOPHANTINE EQUATION.** Felix Klein used
*Diophantische Gleichungen* in
“Die Eindeutigen automorphen Formen vom Geschlechte Null”
in the 1892 issue of *Nachrichten* (page 286):
“Die Relationen kann man in Diophantische Gleichungen umsetzen, welche dann leicht übersehen
lassen, unter welchen Umständen Multiplicatorsysteme möglich sind, und in welcher Anzahl.”
[James A. Landau]

*Diophantine equation* appears in English in 1893 in Eliakim
Hastings Moore (1862-1932), "A Doubly-Infinite System of Simple
Groups," *Bulletin of the New York Mathematical Society,* vol.
III, pp. 73-78, October 13, 1893 [Julio González
Cabillón].

Henry B. Fine writes in *The Number System of Algebra* (1902):

The designation "Diophantine equations," commonly applied to indeterminate equations of the first degree when investigated for integral solutions, is a striking misnomer. Diophantus nowhere considers such equations, and, on the other hand, allows fractional solutions of indeterminate equations of the second degree.

**DIOPHANTINE PROBLEM.** The phrase "Diophantus Problemes" appears
in 1670 [James A. Landau].

The OED2 has the citation in 1700: Gregory, *Collect.* (Oxf.
Hist. Soc.) I. 321: "The resolution of the indetermined arithmetical
or Diophantine problems."

**(DIRAC) δ FUNCTION.**
Paul Dirac
introduced this "improper" function, as he called it, in his book *The Principles of Quantum Mechanics* (1930).
After defining it, Dirac wrote "δ(*x*) is not a function of *x*
according to the usual mathematical definition of a function, ... ,
but is something more general which we may call an 'improper function' to show its
difference from a function defined by the usual definition. ... [Although] an
improper function does not itself have a well-defined value, when it occurs as
a function in an integrand the integral has a well-defined value." See
the entry DISTRIBUTION and the article
Delta-function. [John Aldrich]

**DIRECT VARIATION.** *Directly* is found in English in
Sir Isaac Newton’s *Mathematick Hilosophy [sic] More easily Demonstrated*
by William Whiston: “If a Body be mov’d in an Ellipsis about the Center off the same, the Centripetal Force will be directly as the Distance of the Body from the same Center.”
[Google print search by James A. Landau]

*Directly proportional* is found in 1720 in *Trigonometry Impro’d, and Projection of the Sphere Made Easy* by Henry Wilson:
“The Sines of the Angles subtended by the Perpendiculars, are directly proportional to the Sines of the Perpendiculars subtending.”
[Google print search by James A. Landau]

*Direct variation* is found in 1839 in
*Introduction to the Literature of Europe in the Fifteenth, Sixteenth, and Seventeenth Centuries*
by Henry Hallam [Google print search].

The term **DIRECTION COEFFICIENT** (for cos x + i sin x) is due to
Hankel (1867) (Smith, 1906).

*Direction coefficient* is found in English in a 1902 reprint, 1896 copyright, of a translation of the fourth German edition of *Elements of the Theory of Functions of a Complex Variable*
by H. Durège. The first edition of Durège’s work appeared in 1864 and the second edition in 1873 and the
title was *Theorie der Functionen.* [Google print search by James A. Landau]

Argand used the term *direction factor* (Smith, 1906).

Cauchy used the term *reduced form* (l'expression
réduite) (Smith, 1906).

**DIRECTION FIELD** is found in the context of an alternating current through a coil in 1900 in *The Builder* [Google print search by James A. Landau].

*Direction field* is found in 1910 in "The Theorem of Thomson and Tait and Natural Families of Trajectories"
by Edward Kasner in *Transactions of the American Mathematical Society,* Vol. 11, No. 2 (Apr., 1910), pp. 121-140.
[*JSTOR* serach]

See SLOPE FIELD.

**DIRECTIONAL DERIVATIVE** is found in 1899 in *A Short Table of Integrals* by B. O. Pierce:
“The directional derivative of any scalar point function, *u,* at any point, *P,* in any fixed direction *PQ*’,
is the limit, as *PQ* approaches zero, of the ratio of u_{Q} - u_{P} to *PQ,*
where *Q* is a point on the straight line *PQ*’ between *P* and *Q*’.
[Google print search by James A. Landau]

**DIRECTRIX.** According to the DSB, Jan de Witt (1625-1672) "is
credited with introducing the term 'directrix' for the parabola, but
it is clear from his derivation that he does not use the term for the
fixed line of our focus-directrix definition."

**DIRICHLET INTEGRAL AND DIRICHLET DISTRIBUTION.** Several integrals are associated with
Lejeune Dirichlet: see
MathWorld: Dirichlet Integrals.
One is associated with DIRICHLET’S PRINCIPLE. Another,
considered here, is a multivariable extension of the BETA-FUNCTION,
which was first presented in Dirichlet’s “Sur une nouvelle méthode pour la
détermination des intégrales multiples,”
*Journal de mathématiques pures et appliquées
(Liouville)* **4** 1839) 164-9. The
associated probability distribution, a generalisation of the BETA-DISTRIBUTION,
is given in W. G. Madow’s “Contributions to the Theory of Multivariate
Statistical Analysis,” *Transactions of the
American Mathematical Society*, **44**,
(1938), 454-495. (*JSTOR* search.)
However, it seems that the distribution is much older and even predates
Dirichlet: it appears in a Bayesian analysis in Laplace’s “Mémoire sur les probabilités,”
*Mémoires de l'Académie royale des sciences de Paris*,
1778, pp. 227-332. For this and other appearances of the distribution, see
R. D. Gupta, and D. St. Richards “The History of the Dirichlet and Liouville
Distributions,” *International Statistical
Review*, **69**, (2001),
433-446. See also Wikipedia:
Dirichlet distribution. [John Aldrich]

*Dirichlet’s integral* is found in English in 1851 in “On the index symble of homogeneous functions” by Robert Carmichael in
*The Cambridge and Dublin Mathematical Journal*: “It will be seen that this result bears a strong resemblance to Liouville’s well-known extension of Dirichlet’s Integral.”
[Google print search by James A. Landau]

**DIRICHLET’S PRINCIPLE** and **PROBLEM.** According to G. Birkhoff & E. Kreyszig “The Establishment of
Functional Analysis,” *Historia Mathematica*, **11**, p. 271 the **problem** which involves the solution of a
certain partial differential equation had been posed by Gauss in 1840 and discussed
by W. Thomson in 1847 around the time Dirichlet treated it in his lectures. Dirichlet’s
contribution appeared as “Ueber einen neuen Ausdruck zur Bestimmung der Dichtigkeit
einer unendlich dünnen Kugelschale, wenn der Werth des Potentials derselben in
jedem Punkte ihrer Obertläche gegeben ist,” *Abhandlungen
der Königlich Preussischen Akademie der Wissenchaften von 1850*, S.
99-116. See
*Werke 2* p. 69. Riemann introduced the term **Dirichlet principle** (*Dirichlet’sches Princip*)
for the method of solving the problem by finding the function that minimises a
certain integral and which Dirichlet had described in his lectures: see
Riemann’s “Grundlagen für eine allgemeine Theorie
der Funktionen einer veränderlichen komplexen Grösse” (1851) in
*Werke
1 p. 47*. See the *Encyclopaedia of
Mathematics* entries
Dirichlet problem and
Dirichlet principle.

The English expressions are found in 1893 in *A Treatise on the Theory of Functions* by
James Harkness and Frank Morley where they write, none too accurately, “The
basis for Riemann’s work is a famous proposition known among continental
mathematicians as Dirichlet’s Principle, or Problem.”

**DISCRETE** appears in English in 1570 in Sir Henry Billingsley's
translation of Euclid's *Elements*: "Two contrary kynds of
quantity; quantity discrete or number, and quantity continual or
magnitude" (OED2).

**DISCRETE MATHEMATICS.** The *Twelfth Annual Report of the Ohio
State Academy of Science,* April 15, 1904, has: “The new
mathematics...has triumphed for its own domain in cases where the
continuity methods were wholly inapplicable, where arithmology, discrete
mathematics was called-for and victorious. ... The brilliant and rapid
advances in chemistry have come not from suppressing but from stressing
the individuality of the elements. Its mathematics has been essentially
discrete.” [Google print search by James A. Landau]

In 1931 *Science at the Cross Roads: Papers Presented to the
International Congress of the History of Science and Technology*
has: “We are thus presented with two extremes:
continuity transformed into the absolute and the discrete made absolute,
both laying claim to independent and sole significance, neither seeking
reciprocal unity, each giving a one-sided picture of material reality.
Consequently, the problem is not solved by rejected both analysis and
discrete mathematics....” [Google print search by James A. Landau]

In 1959 “Some Basic Mathematical Concepts,” School Mathematics Study Group has: “What is needed is a discrete mathematics to parallel and supplement our better known continuous mathematics.” [Google print search by James A. Landau]

Classified advertisements appear in 1960 in *The New Scientist.*
It is a help-wanted ad from Sperry Rand Research Center saying there
were openings in the Department of “Applied Mathematics &
Theoretical Physics: Theoretical Physics; Information Theory; Discrete
Mathematics; Random Processes.” No further context is available to
explain what was meant by “discrete mathematics.” [Google
print search by James A. Landau]

*Discrete Mathematics* occurs in 1971 in the title of the
journal
*Discrete Mathematics.*
The fields
covered include: graph and hypergraph theory, network theory, coding
theory, block designs, lattice theory, the theory of partially ordered
sets, combinatorial geometrics, matroid theory, extremal set theory,
logic and automata, matrices, polyhedra, and discrete probability theory.

See FINITE MATHEMATICS.

**DISCRIMINANT.** An English translation of *Geschichte der Elementar-Mathematik* by Karl Fink has:

It was Sylvester also, who in 1851 introduced the name “discriminant” for the function which expresses the condition for the existence of two equal roots of an algebraic equation; up to this time, it was customary, after the example of Gauss, to say “determinant of the function.”

The citation is Sylvester J. J. (1851) “On a remarkable discovery in the theory of canonical
forms and of hyperdeterminants,”
*Phil Mag,* series 2, (1851), 391-410.
[Dick Nickalls]

In 1876 George Salmon used *discriminant* in its modern sense in
*Mod. Higher Algebra* (ed. 3): "The discriminant is equal to the
product of the squares of all the differences of the differences of
any two roots of the equation" (OED2).

The term *geometric discriminant* is described in detail in
Nickalls RWD and Dye RH (1996)
The geometry of the discriminant of a polynomial,
*The Mathematical Gazette* (1996), vol 80 (July), pp 279--285.
The authors believe they are the first to define this geometric analog of the
typical “algebraic” discriminant.

R. A. Fisher introduced and named the **DISCRIMINANT FUNCTION**,
the principal tool of discriminant analysis, in
"The Use of Multiple Measurements in Taxonomic Problems"
*Annals of Eugenics, ***7**: 179-188 (1936). The phrase **DISCRIMINANT
ANALYSIS** appears in Fisher’s second paper
on the subject
"The Statistical Utilization of Multiple Measurements"
*Annals of Eugenics,* **8** : 376-386 (1938). (David 2001)

**DISJOINT** is found in 1909 in C. J. Keyser "The Thesis of Modern Logistic,"
*Science*, New Series, **30**, No. 783. (Dec. 31), p. 956: "two classes are *disjoint*
if neither includes a term of the other." (*JSTOR*)

The converse, *conjoint*, was *not* applied to classes
but it was used elsewhere: from George Ballard Mathews
*Projective Geometry* (1914), "Taking any two elements of different names
(plane, point; plane, line; point, line), we may distinguish them as being *disjoint* or *conjoint* (p. 3).

**DISJUNCTION** was considered by the Stoic
logicians. W. & M. Kneale *The Development of Logic* (1962) pp. 160-1
state that when it came to interpretations of disjuction
Chrysippus of Soli
(280 BC - 206 BC) and his followers "seem to have assumed" that exclusive disjunction, i.e.
that two alternatives could not be true together, "was the only proper
one." For general information on the Stoics see Dick Baltzly
Stoicism.

*Disjunction* and *disjunctive* came into
English in the 16^{th} century, not in textbooks on logic, which continued
to be written in Latin for another century, but in works for other audiences.
The OED's earliest quotations come from Abraham Fraunce *The Lawiers
Logike, exemplifying the Praecepts of Logike by the Practise of the Common Lawe*
(1588) and from Dudley Fenner *A Defence of the Godlie Ministers* (1587),
a book on church government. The word is traced to Old French and to Latin.

**DISME** is an obsolete English word meaning "tenth." It occurs
in 1608 in the title *Disme: The Art of Tenths, or Decimall
Arithmetike.* This work is a translation by Robert Norman of *La
Thiende,* by Simon Stevin (1548-1620), which was published in
Flemish and in French in 1585.

*Disme* was used by Shakespeare in *Troilus and Cressida*
(ii, 2, 15), which was first published in 1609. The use of this word
is one of the pieces of evidence cited by defenders of the theory
that Shakespeare's plays were actually written someone else, perhaps
Francis Bacon.

**DISPERSION** (in statistics) is found in 1876 in *Catalogue of
the Special Loan Collection of Scientific Apparatus at the South
Kensington Museum* by Francis Galton (David, 1998).

**DISTRIBUTED LAG.** The term was used by Irving Fisher in his 1925 paper "Our Unstable Dollar and the
so-called Business Cycle" (*Journal of the American Economic Association,* **20,** 179-202) to describe
a formulation he had used in "The Business Cycle Largely a 'Dance of the Dollar,'" *Journal of the American
Statistical Society,* **18,** (1923), 1024-1028 [John Aldrich].

See the entry LAG.

**DISTRIBUTION **in the sense of a generalized function was introduced by
Laurent Schwartz in his paper
*Généralisation
de la notion de fonction, de dérivation, de transformation de Fourier et applications mathématiques et physiques* (1945).
This presented a theory of distributions in which such "improper functions"
as the DIRAC δ FUNCTION could be treated rigorously. See the article
Generalized function.

The term **DISTRIBUTION FUNCTION** of a random variable
is a translation of the *Verteilungsfunktion* of R. von Mises "Grundlagen der Wahrscheinlichkeitsrechnung,"
*Math.
Zeit.* **5**, (1919), 52-99.

The English term appears in J. L. Doob's "The Limiting
Distributions of Certain Statistics," *Annals of Mathematical Statistics,*
**6**, (1935), 160-169.

The term *cumulative distribution function* was used
by S. S. Wilks *Mathematical Statistics* (1943) (David 2001).

(See also Symbols in Probability on the Symbols in Probability and Statistics page.)

**DISTRIBUTIVE.** See *commutative.*

The term **DIVERGENCE** (of a vector field) was introduced
by William Kingdon Clifford (1845-1879) in his book *Elements of Dynamic*
(1878). It was the negative of the quantity James Clerk Maxwell had called the
*convergence*. Maxwell suggested this term in 1870 in a letter to Tait,
reproduced in the entry CURL. It was proposed more formally in Maxwell’s
"On the Mathematical Classification of Physical Quantities," *Proceedings
of the London Mathematical Society*, **3**, (1871), 257-266. Maxwell
gave a fuller account of the term in the section, *Effect of Hamilton’s operation*
*on a vector function*, of his
*A Treatise on
Electricity and Magnetism* (1873, p. 27)

Oliver Heaviside started using the term divergence in 1883 and he may have come up with the term independently of Clifford.

See also QUATERNION and VECTOR ANALYSIS
for related terms and **Vector Calculus Symbols** on the the
Earliest Uses of Symbols of Calculus page.

[Based on M. J. Crowe *A History of Vector Analysis*
(2^{nd} edition, pp. 131-2, 165-6)]

**DIVERGENCE THEOREM** or **GAUSS'S THEOREM.** A *JSTOR* search found *Gauss’s theorem* in Henry A. Rowland
“On the Motion of a Perfect Incompressible Fluid When no Solid Bodies are
Present,” *American Journal of Mathematics*, **3**, (1880),
pp. 226-268 and *divergence theorem* in Oliver Heaviside “On the Forces, Stresses, and Fluxes of Energy in the
Electromagnetic Field,” *Philosophical
Transactions of the Royal Society A,* **183** (1892), 423-480.
It seems likely however that both terms were already established.

The history of the theorem is bewildering with many re-discoveries.

O. D. Kellogg *Foundations of Potential Theory* (1929, p. 38) has the following note on the result “known as the Divergence
Theorem, or as Gauss’ Theorem or Green’s Theorem”:

A similar reduction of triple integrals to double integrals was employed by Lagrange:Nouvelles recherches sur la nature et la propagation du sonMiscellanea Taurinensis, t. II, 1760-61; Oeuvres t. I, p. 263. The double integrals are given in more definite form by GaussTheoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo novo tractate, Commentationes societas scientiarum Gottingensis recentiores, Vol III, 1813,WerkeBd. V pp. 5-7. A systematic use of integral identities equivalent to the divergence theorem was made by George Green in hisEssay on the Mathematical Theory of Electricity and Magnetism; Nottingham, 1828 [Green Papers, pp. 1-115].

Kline (pp. 789-90) writes that
Mikhail
Ostrogradski obtained the theorem when solving the partial differential
equation of heat. He published the result in 1831 in *Mem. Ac. Sci. St. Peters*., **6**,
(1831) p. 39. J. C. Maxwell had made the same attribution in the 2^{nd} edition of the *Treatise on Electricity and
Magnetism* (1881). See also the
Encyclopaedia of Mathematics entry
Ostrogradski formula

For Gauss’s theorem Hermann Rothe “Systeme Geometrischen Analyse, Erster Teil”
Encyklopädie
der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen
Volume: 3, T.1, H.2 p. 1345 refers to Gauss’s *Allgemeine Lehrsätze in Beziehung auf die im
verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossung-Kräfte* 1839 in
*Werke* Bd. V (especially pp. 226-8.)

[This entry was contributed by John Aldrich, and was based on M. J. Crowe *A History of Vector Analysis* (2^{nd} edition, pp. 146-7)]

**DIVERGENT.** See *convergent.*

**DIVIDEND.** Joannes de Muris (c. 1350) used *dividendus*
(Smith vol. 2, page 131).

In English, the word is found in *The Grovnd of Artes,* by
Robert Recorde, which was printed between 1540 and 1542: "Then
begynne I at the hyghest lyne of the diuident, and seke how often I
may haue the diuisor therin" (OED2).

The term **DIVINE PROPORTION** appears in 1509 in the title *De
Divina Proportione* by Luca Pacioli (1445-1517). According to an
Internet website, Pacioli coined the term.

Ramus wrote, "Christianis quibusdam divina quaedam proportio hic
animadversa est..." in *Scholarvm Mathematicarvm, Libri vnvs et
triginta,* Basel, 1569; *ibid.,* 1578; Frankfort, 1599)
(Smith vol. 2, page 291).

Kepler wrote, "Inter continuas proportiones unum singulare genus est
proportionis divinae" (Frisch ed. of his *Opera,* I (1858).
According to v. Baravalle (1948), Kepler used the term *sectio
divina.*

**DIVISION** is found in English in "The crafte of nombrynge" (ca.
1300). The word is spelled *dyuision* (OED2).

Baker (1568) speaks of "Deuision or partition" and Digges (1572) says "To deuide or parte" (Smith vol. 2, page 129).

**DIVISOR** is found in English in "The crafte of nombrynge" (ca.
1300). The word is spelled *dyvyser* (OED2).

**DODECAGON** is found in English in a mathematical dictionary
of 1658.

**DOMAIN** was used in 1886 by Arthur Cayley in
"On Linear Differential Equations" in
the *Quarterly Journal of Pure and Applied Mathematics*:
"... for points *x* within the domain of the point *a*
[University of Michigan Historic Math Collection].

*Domain* is found in 1896 in *Bull. Amer. Math. Soc.* Dec.
103: "The formation of an algebraic 'domain' and..the nature of the
process of 'adjunction' introduced by Galois" (OED2).

*Domain,* referring to a power series, appears in 1898 in
*Introduction to the theory of analytic functions* by J.
Harkness and F. Morley.

*Domain,* in the sense of the values that an independent
variable of a function can take, appears in the *Encyclopaedia
Britannica* of 1902 (OED2).

**DOMINANCE** (in game theory and decision theory). The term "dominates" is prominent in
J. von Neumann and O. Morgenstern *Theory of Games and Economic Behavior* (1944).
L. J. Savage *Foundations of Statistics*
(1954) used the term in statistical decision theory. Wald,
the founder of that theory, had used "absolutely better than," e.g. in his
"Contributions to the Theory of Statistical Estimation and Testing Hypotheses,"
*Annals of Mathematical Statistics,* **10**, (1939), 299-326.

**DOPPELVERHÄLTNISS.** Möbius introduced the term
*Doppelschnittverhältniss,* meaning "ratio bisectionalis"
or "double cut ratio," in his "Der barycentrische Calcul" (1827):
gesammelte Werke, I (1885).

Jakob Steiner shortened the term to *Doppelverhältniss*
(Smith vol. 2, page 334).

See also *anharmonic ratio* and *cross-ratio.*

**DOT PRODUCT** is found in 1901 in *Vector Analysis* by J.
Willard Gibbs and Edwin Bidwell Wilson:

The direct product is denoted by writing the two vectors with a dot between them as[This citation was provided by Joanne M. Despres of Merriam-Webster Inc.]

A·BThis is read

AdotBand therefore may often be called the dot product instead of the direct product.

**DOUBLE INTEGRAL** appears in the title of Adrien-Marie Legendre,
“Mémoire sur les Intégrales Doubles,” Paris, Mem. Acad. Sc. for 1788,
published 1791, pp. 454-486. [James A. Landau]

**DUALITY.** The term "principle of duality" was introduced by
Joseph Diaz Geronne (1771-1859) in "Considérations
philosophiques sur les élémens de la science de
l'étendue," *Annales* 16 (1825-1826) (DSB).

The term **DUMMY VARIABLE** is often used when describing
the status of a variable like *x* in a definite integral. A. Church seems
to be describing an established usage when he wrote in 1942, "A variable
is free in a given expression ... if the expression can be considered as representing
a function with that variable as an argument. In the contrary case the variable
is called a *bound* (or *apparent* or *dummy*) variable."
("Differentials", *American Mathematical Monthly,* **49**,
390.) [John Aldrich].

In regression analysis a **DUMMY VARIABLE** indicates the
presence (value 1) or absence of an attribute (0).

A JSTOR search found "dummy variables" for social
class and for region in H. S. Houthakker's "The Econometrics of Family
Budgets" *Journal of the Royal Statistical Society A,* **115**,
(1952), 1-28.

A 1957 article by D. B. Suits, "Use of Dummy Variables
in Regression Equations" *Journal of the American Statistical Association,*
**52**, 548-551, consolidated both the device and the name.

Kendall & Buckland in their* Dictionary of Statistical
Terms* object: the term is "used, rather laxly, to denote an artificial
variable expressing qualitative characteristics .... [The] word 'dummy' should
be avoided." Apparently these variables were not dummy enough. For Kendall
& Buckland a dummy variable signifies "a quantity written in a mathematical
expression in the form of a variable although it represents a constant",
e.g. when the constant in the regression equation is represented as a coefficient
times a variable that is always unity.

The indicator device, without the name "dummy variable"
or any other, was also used by writers on experiments who put the analysis of
variance into the format of the general linear hypothesis, e.g. O. Kempthorne
in his *Design and Analysis of Experiments* (1952) [John Aldrich].

**DUODECIMAL** is dated 1663 in MWCD10.

*Duodecimal* appears in 1714 in the title *A new and complete
Treatise of the Doctrine of Fractions .. with an Epitome of
Duodecimals* by Samuel Cunn (OED2).

**DURBIN-WATSON TEST.** The test and associated distribution
theory were given by J. Durbin; G. S. Watson "Testing for Serial Correlation in Least Squares
Regression," Parts I and II, *Biometrika*, 1950-1. In an
*ET*
interview Durbin described the background to the work.

**DUTCH BOOK (ARGUMENT)** in subjective probability theory. The
term entered circulation in the 1950s. R. Sherman Lehman
explains the term, "If a bettor is quite foolish in the choice of the rates
at which he will bet, an opponent can win money from him no matter what happens.
The phenomenon is well known to professional bettors--especially bookmakers ...
Such a losing book is known by them as a ‘dutch book.’ Our investigations are
thus concerned with necessary and sufficient conditions that a book not be ‘dutch.’"
("On Confirmation and Rational Betting," *Journal of Symbolic Logic*,
**20**, (1955), p. 251.)

The classic use of the Dutch book argument to obtain the properties of subjective odds is B. De Finetti’s
"La
prévision: ses lois logiques, ses sources subjectives,"
*Annales de l'Institute Henri Poincaré*, **7**, (1937) 1-68. The term "Dutch book," however, is not used.

**DYAD.** The Greek word *dyad* is found in Euclid in
Proposition 36 of Book IX and in Nichomachus' Introduction to
Arithmetic (Book I, Chapter VII) [Mary Townsend].

In the sense of a mathematical operator, *dyad* was used in 1884
by Josiah Willard Gibbs (1839-1903) and is found in his
*Vector-Analysis* of 1901 and his *Collected Works* of 1928
(OED2).