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

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).

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.

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

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 THEORY (as a term for the approach to statistical inference), inaugurated 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), came into use around 1950. The phrase appears in Lehmann’s "Some Principles of the Theory of Testing Hypotheses," Annals of Mathematical Statistics, 21, (1950), 1-26 [John Aldrich].

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

Augustin Louis Cauchy (1789-1857) used the term. It is found in his 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 for x = a the 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 the definite integral, as it is then called, is Fx + A - Fa.

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. (See also chi-squared, F-distribution and Student's t-distribution.) Fisher introduced degrees of freedom in connection with Pearson's test in the 1922 paper "On the Interpretation of from "Contingency Tables, and the Calculation of P," J. Royal Statist. Soc., 85, pp. 87-94. He applied the number of degrees of freedom to distributions related to chi-squared--Student's distribution and his own z distribution in his 1924 paper, "On a Distribution Yielding the Error Functions of Several well Known Statistics," Proceedings of the International Congress of Mathematics, Toronto, 2, 805-813 [John Aldrich].

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 monosyllable del is 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. 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.

Theorem of De Moivre appears in 1840 in Mathematical Dissertations, for the use of Students in the Modern Analysis, with Improvements in the Practice of Sturm's Theorem, in the Theory of Curvature, and in the Summation of Infinite Series (1841) by J. R. Young [James A. Landau].

Demoivre's formula appears in English in 1849 in An Introduction to the Differential and Integral Calculus, 2nd ed., by James Thomson.

Demoivre's theorem appears in 1859(?) in Plane Trigonometry by the Right Rev. J.W. Colenso [University of Michigan Historic Math Collection].

Demoivre's theorem appears in English in the third edition of An Elementary Treatise on the Theory of Equations (1875) by Isaac Todhunter.

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 VARIABLE. Subordinate variable appears 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].

DERIVATIVE and DIFFERENTIAL COEFFICIENT. The term derivative was not used by Isaac Newton, who instead used the term fluxion.

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

Aequationem differentialem voco talem qua valor ipsius dx exprimitur, quaeque ex alia derivata est, qua valor ipsius x exprimebatur [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, ...

... 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 ...]

An 1816 English translation of Lacroix has: "The limit of the ratio of the increments, or the differential coefficient, will be obtained" (OED2).

In 1891 Differential and Integral Calculus by George A. Osborne prefers the term differential coefficient but has: "The differential coefficient is sometimes called the derivative."

DESCARTES' RULE OF SIGNS. In A Compendious Tract on the Theory of Solutions of Cubic and Biquadratic Equations, and of Equations of the Higher Orders (1833) by Rev. B. Bridge, the rule is called Des Cartes' rule.

In the second edition of Theory and Solution of Algebraical Equations (1843), J. R. Young refers to "the rule of signs."

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).

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:

Numerum bb - ac, a cuius indole proprietates formae (a, b, c) imprimis pendere in sequentibus docebimus, determinantem huius formae uocabimus.

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

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 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:

M. Gauss s'en est servi avec avantage dans ses Recherches analytiques pour 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 de determinants. 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, determinant et resultante, 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) 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.

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 (see below).

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).

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."

DIFFERENTIAL EQUATION. Gottfried Wilhelm Leibniz (1646-1716) introduced the term in 1676, according to Franceschetti (p. 401).

Leibniz 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).

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 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).

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.

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

DIRECT VARIATION. Directly is found in 1743 in W. Emerson, Doctrine Fluxions: "The Times of describing any Spaces uniformly are as the Spaces directly, and the Velocities reciprocally" (OED2).

Directly proportional is found in 1796 in A Mathematical and Philosophical Dictionary: "Quantities are said to be directly proportional, when the proportion is according to the order of the terms" (OED2).

Direct 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.

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

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

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

DIRECTIONAL DERIVATIVE is found in 1912 in Advanced Calculus by Edwin Bidwell Wilson: "The derivative (13) is called the directional derivative of f in the direction of the line."

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'S PROBLEM is found in 1893 in A Treatise on the Theory of Functions by James Harkness and Frank Morley: "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 occurs in 1971 in the title of the journal Discrete Mathematics.

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).

DISCRIMINANT ANALYSIS is found in Palmer O. Johnson, "The quantification of qualitative data in discriminant analysis," J. Am. Stat. Assoc. 45, 65-76 (1950).

See also W. G. Cochran and C. I. Bliss, "Discriminant functions with covariance," Ann. Math. Statist. 19 (1948) [James A. Landau].

DISJOINT is found in 1914 in Projective Geometry by George Ballard Mathews: "Taking any two elements of different names (plane, point; plane, line; point, line), we may distinguish them as being disjoint or conjoint [University of Michigan Historic Math Collection].

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.

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].

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).

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). 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).

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

A·B

This is read A dot B and therefore may often be called the dot product instead of the direct product.

DOUBLE INTEGRATION appears in 1831 in Elements of the Integral Calculus (1839) by J. R. Young.

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.

The International Statistical Institute's Dictionary of Statistical Terms objects to the name: 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, for whom 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).

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).