**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."

*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.

The word **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].

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

**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)** is found in 1730-6, Bailey (folio):
"*Parodic Degree* (in Algebra) is the index or exponent of any power;
so in numbers, 1. is the parodick degree, or exponent of the root or side;
2. of the square, 3. of the cube, etc" (OED2).

**DEGREE OF FREEDOM** (in statistics) is found in R. A. Fisher, *Jrnl.
R. Statistical Soc.* LXXXV 88 (OED2).

**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).

The term **DELTAHEDRON** was coined by H. Martyn Cundy. The word
may occur in *Mathematical Models* (1961), by him and A. P. Rollett.

**DEMOIVRE'S THEOREM** appears in the *Century Dictionary* (1889-1897).

**DEPENDENT VARIABLE** is found in 1852 in *Differential Calculus*
by Isaac Todhunter (OED2).

**DERIVATIVE.** This term was not used by Isaac Newton, who instead
used the term *fluxion.*

Julio González Cabillón believes that this word was first used in the
calculus sense (in Latin, *derivata*) by Gottfried Wilhelm Leibniz
(1646-1716) around 1677:

Aequationem differentialem voco talem qua valor ipsiusJohn Conway points out that, in the above, it could be argued that Leibniz is merely using the word "derived" in its ordinary sense.dxexprimitur, quaeque ex alia derivata est, qua valor ipsiusxexprimebatur [cf. page 156 of Leibniz' "Mathematische Schriften," vol. I, edited by C. I. Gerhardt, Verlag von A. Asher & Comp., Berlin, 1849].

Some writers attribute the word *derivative* to Joseph Louis Lagrange
(1736-1813), who 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.*
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, ...The term... 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 ...]

**DETERMINANT (discriminant of a quantic)** was introduced in 1801
by Carl Friedrich Gauss in his *Disquisitiones arithmeticae:*

Numerum(Cajori vol. 2, page 88; Smith vol. 2, page 476).bb-ac,a cuius indole proprietates formae (a, b, c) imprimis pendere in sequentibus docebimus,determinantemhuius formae uocabimus.

**DETERMINANT (modern sense).** Augustin-Louis Cauchy (1789-1857)
was apparently the first to use *determinant* in its modern sense
(Schwartzman, page 70). He employed the word in "Memoire sur les foctions
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:

M. Gauss s'en est servi avec avantage dans ses(Smith vol. 2, page 477; Julio González Cabillón.)Recherches 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.

According to Katz, Arthur Cayley (1821-1895) introduced the word *determinant*
as a replacement for several older terms.

**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) to refer
to a method of eliminating a variable from two algebraic equations (Schwartzman,
page 71).

**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."

**DIFFERENTIABLE.** Leibniz used the term *differentiabilis*
in 1692 in *Acta Eruditorum 11* (Struik, page 272).

The noun **DIFFERENTIAL** was coined by Leibniz, according to an
Internet website. The word appears in English as a noun in 1704 in *Lexicon
technicum, or an universal English dictionary of arts and sciences*
by John Harris (OED2).

**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).

Leibniz wrote [source uncertain]: "Knowing thus the Algorithm of this calculus, which I call Differential Calculus, all differential equations can be solved by a common method."

The term **DIFFERENTIAL COEFFICIENT** was first used by Sylvestre-François
Lacroix (1765-1843) in *Traité du calcul différentiel et integral*
(Cajori 1919, page 272).

**DIFFERENTIAL EQUATION.** Gottfried Wilhelm Leibniz (1646-1716)
used the Latin *aequationes differentiales* in *Acta Eruditorum,*
October 1684. See the entry "algorithm" for the context.

The term **DIFFERENTIAL GEOMETRY** was first used by Luigi Bianchi
(1856-1928) in 1894 (Kline, page 554).

**DIFFERENTIATE** appears in English in 1816 in *LaCroix's Differential
and Integral Calculus* (OED2).

**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).

**DIOPHANTINE ANALYSIS** appears in 1811 in the title *An Elementary
Investigation of the Theory of Numbers, with its application to the indeterminate
and diophantine analysis* by Peter Barlow (OED2).

An early use of the term **DIOPHANTINE EQUATION** in English is by
Eliakim Hastings Moore (1862-1932) in an essay entitled "A Doubly-Infinite
System of Simple Groups," published in the *Bulletin of the New York
Mathematical Society,* vol. III, pp. 73-78, October 13, 1893 [Julio
González Cabillón].

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

**DIRECT VARIATION** is found in 1881 in *Elements of Algebra*
by G. A. Wentworth [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."

**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).

**DISCRIMINANT** was introduced by James Joseph Sylvester (1814-1897)
in 1852 in the *Cambridge and Dublin Mathematical Journal,* vol. I,
52. He used the word "for *determinant,* which is still found occasionally,"
according to the OED2, which attributes this information to H. T. Gerrans.

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).

**DISJOINT,** referring to intervals, occurs in "Differentiation
with Respect to a Function of Limited Variation," P. J. Daniell, *Transactions
of the American Mathematical Society,* Vol. 19, No. 4. (Oct., 1918).

*Disjoint,* referring to sets, is found in the phrase "two disjoint
closed sets" in 1937 in *Transactions of the American Mathematical Society*
(OED2).

**DISJUNCTION.** According to the University of St. Andrews website,
"the logical term 'disjunction' is certainly due to the Stoics and it is
thought to have originated with" Chrysippus of Soli (280 BC - 206 BC).

**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.

**DISTRIBUTIVE.** See *commutative.*

The term **DIVERGENCE** (of a vector field) was introduced by William
Kingdon Clifford (1845-1879). Maxwell had earlier used the term *convergence*
with a related meaning (Kline, page 785).

The DSB says that Maxwell introduced the term *divergence* in 1870;
this seems to be incorrect.

**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).

**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 dated ca. 1658 in MWCD10.

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

**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.

**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).

**DUMMY VARIABLE** is dated 1957 in MWCD10.

**DYAD** (a third vector product) 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).