# Earliest Known Uses of Some of the Words of Mathematics (V)

**VANDERMONDE DETERMINANT, MATRIX etc.**It seems that the determinant named after Alexandre-Theophile Vandermonde (1735-96) was never discussed by him. Lebesgue believed that the attribution was due to a misreading of Vandermonde's unfamiliar notation. (DSB and H Lebesgue, "L'oeuvre mathématique de Vandermonde,"

*Enseignement Math.*(2)

**1**(1956), 203-223.). It is not known who coined the term but it appears in 1888 in G. Weill, "Sur une forme du déterminant de Vandermonde,"

*Nouv. Ann.*(1888).

The term

**VANISHING POINT**was coined by Brook Taylor (1685-1731), according to Franceschetti (p. 500).

The term

**VARIABLE**was introduced by Gottfried Wilhelm Leibniz (1646-1716) (Kline, page 340).

*Variable*is found in English as an adjective in 1710 in

*Lexicon Technicum*by J. Harris: "

*Variable Quantities,*in Fluxions, are such as are supposed to be continually increasing or decreasing; and so do by the motion of their said Increase or Decrease Generate Lines, Areas or Solidities" (OED2).

*Variable*is found in English as a noun in 1816 in a translation of

*Lacroix's Differential and Integral Calculus*: "The limit of the ratio..will be obtained by dividing the differential of the function by that of the variable" (OED2).

**VARIANCE, ANALYSIS OF VARIANCE,**and

**ANOVA.**The term

*variance*was introduced by Ronald Aylmer Fisher in 1918 in a paper on population genetics, The Correlation Between Relatives on the Supposition of Mendelian Inheritance

*Transactions of the Royal Society of Edinburgh*,

**52**, 399-433: "It is ... desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. We shall term this quantity the Variance ..." (p. 399)

The phrase

*analysis of variance*appears in the table of contents of the same paper, but not in the text. It appears in the text of a non-technical exposition Fisher gave, "The Causes of Human Variability,"

*Eugenics Review*,

**10**, (1918), 213-220. In these pieces the analysis of variance refers to a decomposition of the population variance based on genetical theory.

The statistical procedure which Fisher also called the analysis of variance--and which is universally known by that name--came a bit later. In a 1921 paper ( Studies in Crop Variation. I. An examination of the yield of dressed grain from Broadbalk

*Journal of Agricultural Science*,

**11**, 107-135) Fisher discussed "the analysis of the total variance for each plot." (p. 111) The analysis of variance table first appeared in a note written by Fisher contained in Student's 1923 paper "On Testing Varieties of Cereals,"

*Biometrika*,

**15**, p. 283 footnote. Fisher's book

*Statistical Methods for Research Workers*(1925) made the analysis of variance widely known.

The later literature found

*two*models in Fisher's statistical work and produced contrasting terms for them. C. Eisenhart in "The Assumptions Underlying the Analysis of Variance,"

*Biometrics*,

**3**, (1947), 1-21 called them

**Model I**and

**Model II**analysis of variance. Eisenhart also used the terms

**fixed effects**and

**random effects**, though more informally. These appear with greater ceremony in H. Scheffé's "Alternative Models for the Analysis of Variance,"

*Annals of Mathematical Statistics*,

**27**, (1956), 251-271.

**Components of variance**, another common term, was introduced by H. E. Daniels in 1939 in his "The Estimation of Components of Variance,"

*Supplement to the Journal of the Royal Statistical Society*,

**6**, 186-197.

The acronym

*ANOVA*is used in a 1949 article by John W. Tukey: "Dyadic anova: an analysis of variance for vectors" (

*Human Biology*,

**21**, 65-110).

See also ERROR, F-DISTRIBUTION, HYPOTHESIS and HYPOTHESIS-TESTING, STANDARD DEVIATION, z-DISTRIBUTION.

[This entry was contributed by John Aldrich, based on OED2, David (1995) and J. F. Box (1978)

*R. A. Fisher: The Life of a Scientist*, New York, Wiley.]

**VARIATE**appears in 1909 in Karl Pearson, "On a New Method of Determining Correlation...,"

*Biometrika,*7, 96-105 (David, 1998).

Pearson and some later statisticians, including R. A. Fisher, used the term in the sense of the modern

*random variable*: "The variable quantity, such as the number of children, is called the

**variate**, and the frequency distribution specifies how frequently the variate takes each of its possible values."

*Statistical Methods for Research Workers*(1925, p. 4.) (OED2)

See RANDOM VARIABLE.

**VARIATE DIFFERENCE (CORRELATION) METHOD**appears in the title of Beatrice M. Cave and Karl Pearson's "Numerical Illustrations of the Variate Difference Correlation Method,"

*Biometrika*,

**10**, (1914), 340-355. (David (2001).

The name refers to a method whose history Cave and Pearson gave as follows, "In 1904 Miss F. E. Cave in a memoir on the correlation of barometric heights published in the

*R.S. Proc*, Vol. LXXIV [p. 403 On the Influence of the Time Factor on the Correlation between the Barometric Heights at Stations More than 1000 Miles apart] endeavoured to get rid of seasonal change by correlating first differences of daily readings at two stations. ... This method was generalised by Student in the last issue of

*Biometrika*["The Elimination of Spurious Correlation due to Position in Time or Space,"

*Biometrika,*

**10,**(1914), 179-180].

When Gerhard Tintner (

*Variate Difference Method*(1940)) suggested the use of differencing when calculating the variance of the error around the trend of a single series the term

*correlation*was no longer appropriate and was dropped.

See also SPURIOUS CORRELATION.

**VARIETY**(as in modern algebraic geometry) was first used by E. Beltrami in 1869 [Joseph Rotman].

Birkhoff used the term

*equationally defined algebras*in his AMS Colloquium Volume

*Lattice Theory*in the first 1940, second 1948 and third 1967 edition.

Hanna Neumann (1914-1971) introduced the term

*variety*in "On varieties of groups and their associated near-rings,"

*Math. Zeits.,*65, 36-69 (1956) and popularised the term in her 1967 book

*Varieties of Groups*[Phill Schultz].

**VECTOR, VECTOR ANALYSIS**and

**VECTOR SPACE.**The entries below follow changes in the use of the term

*vector*over the past 150 years or so. The story begins with a technical term in theoretical astronomy, "radius vector," in which "vector" signified "carrier." In the first and biggest change "radius" was dropped and "vector" was given a place in the algebra of quaternions (c. 1840). The recognition that vectors are more interesting than quaternions, especially in physics, led to

*vector analysis*(c. 1880). In the 20

^{th}century

*vector*was extended from triples of numbers to

*n*-tuples and then to elements of abstract linear spaces.

The word

**VECTOR**(which, like the word

*vehicle*, derives ultimately from the Latin

*vehĕre*to carry) was first a technical term in astronomical geometry. The OED's earliest entry is from a technical dictionary of 1704: J. Harris

*Lexicon Technicum*I. s.v., "A Line supposed to be drawn from any Planet moving round a Center, or the Focus of an Ellipsis, to that Center or Focus, is by some Writers of the New Astronomy, called the Vector; because 'tis that Line by which the Planet seems to be carried round its Center."

Vector usually appeared in the phrase radius vector. The French term was

*rayon vecteur*and can be found in e.g. Laplace's

*Traité de mécanique céleste*(1799-1825).

The term

*rayon vecteur*is used in a non-astronomical context by Monge "Application de l'Analyse à la Géométrie" (1807). On p. 24 he writes, "on nomme la droite r le RAYON VECTEUR du point, et l'origine des coordonnées devient un pôle, d'où partent les rayons vecteurs des différens points de l'espace." Monge does not seem to use the terminology later in the text, however. Cauchy in his

*Leçons sur les Applications du Calcul Infinitésimal à la Géométrie*of 1826 uses the term freely after introducing it in the initial "Preliminaries" chapter: "Une droite AB, menée d'un point A supposé fixe à un point B suppose mobile, sera généralement désignée sous le nom de

*rayon vecteur*." (p.14) (Information from François Ziegler.) Radius vector appears in English in 1831 in

*Elements of the Integral Calculus*(1839) by J. R. Young: "...when the angle Ω between the radius vector and fixed axis is taken for the independent variable, the formula is...."

Hamilton would create a new meaning for

*vector*(see VECTOR & SCALAR) but he used

*radius vector*in the conventional way in On a General Method in Dynamics Philosophical Transactions Royal Society (part II for 1834, pp. 247-308); see article 14.

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

**VECTOR**and

**SCALAR**. Both the terms

*vector*and

*scalar*were introduced by William Rowan Hamilton (1805-1865).

Both terms appear in "On quaternions" a paper presented by Hamilton at a meeting of the Royal Irish Academy on November 11, 1844. This paper adopts the convention of denoting a vector by a single (Greek) letter, and concludes with a discussion of formulae for applying rotations to vectors by conjugating with unit quaternions. It is on pages 1-16 in volume 3 of the

*Proceedings of the Royal Irish Academy,*covering the years 1844-1847, and the volume is dated 1847. The following is from page 3:

On account of the facility with which this so called imaginary expression, or square root of a negative quantity, is constructed by a right line having direction in space, and havingThe following is from page 8:x, y, zfor its three rectangular axes, he has been induced to call the trinomial expression itself, as well as the line which it represents, a VECTOR. A quaternion may thus be said to consist generally of a real part and a vector. The fixing a special attention on this last part, or element, of a quaternion, by giving it a special name, and denoting it in many calculations by a single and special sign, appears to the author to have been an improvement in his method of dealing with the subject: although the general notion of treating the constituents of the imaginary part as coordinates had occurred to him in his first researches.

It is, however, a peculiarity of the calculus of quaternions, at least as lately modified by the author, and one which seems to him important, that it selects no one direction in space as eminent above another, but treats them as all equally related to that extra-spacial, or simply SCALAR direction, which has been recently called "Forward."In Hamilton's time

*radius-vector*was an established term in astronomy (see previous entry). Hamilton explains that he is giving the term

*vector*a new sense in Lecture I of his

*Lectures on Quaternions*. In article 17 (p. 16) he described the difference between vector and radius-vector:

17. To illustrate more fully the distinction which was just now briefly mentioned, between the meanings of the "Vector" and the "Radius Vector" of a point, we may remark that the RADIUS-VECTOR, in astronomy, and indeed in geometry also, is usually understood to have only length; and therefore to be adequately expressed by a SINGLE NUMBER, denoting the magnitude (or length) of the straight line which is referred to by this usual name (radius-vector) as compared with the magnitude of some standard line, which has been assumed as the unit of length. Thus, in astronomy, the Geocentric Radius-Vector of the Sun is, in its mean value, nearly equal to ninety-five millions of miles: if, then, a million of miles be assumed as the standard or unit of length, the sun's geocentric radius-vector is equal (nearly) to, or is (approximately) expressible by, the number ninety-five: in such a manner that this single number, 95, with the unit here supposed, is (at certain seasons of the year) a full, complete and adequate representation or expression for that known radius vector of the sun. For it is usually the sun itself (or more fully the position of the sun's centre) and NOT the Sun's radius-vector, which is regarded as possessing also certain other (polar) coordinates of its own, namely, in general, some two angles, such as those which are called the Sun's geocentric right-ascension and declination; and which are merely associated with the radius-vector, but not inherent therein, nor belonging thereto...Hamilton, in his "Lectures on Quaternions", is not satisfied with having introducedBut in the new mode of speaking which it is here proposed to introduce, and which is guarded from confusion with the older mode by the omission of the word "RADIUS," the VECTOR of the sun HAS (itself) DIRECTION, as well as length. It is, therefore NOT sufficiently characterized by ANY SINGLE NUMBER, such as 95 (were this even otherwise rigorous); but REQUIRES, for its COMPLETE NUMERICAL EXPRESSION, a SYSTEM OF THREE NUMBERS; such as the usual and well-known rectangular or polar co-ordinates of the Sun or other body or point whose place is to be examined...

A VECTOR is thus (as you will afterwards more clearly see) a sort of NATURAL TRIPLET (suggested by Geometry): and accordingly we shall find that QUATERNIONS offer an easy mode of symbolically representing every vector by a TRINOMIAL FORM (ix+jy+kz); which form brings the conception and expression of such a vector into the closest possible connexions with Cartesian and rectangular co-coordinates.

*vector.*Within a few pages we find

*vectum, vehend, revector, provector, provectum, transvehend, transvectum,*etc., and identities such as

*Vector*and

*scalar*also appear in 1846 in a paper "On Symbolical Geometry," in the

*The Cambridge and Dublin Mathematical Journal*vol. I:

If then we give the name ofNext Hamilton goes on to tell us about another "chief class" of the "geometrical quotients," namelyscalarsto all numbers of the kind called usually real, because they are all contained on the one scale of progression of number from negative to positive infinity [...]

the class in which the dividend is a line perpendicular to the divisor. A quotient of this latter class we shall call aDavid Wilkins believes that the paper "On quaternions" in thevector,to mark its connection (which is closer than that of a scalar) with the conception of space [...]

*Proceedings of the Royal Irish Academy*probably appeared earlier than the

*CDMJ,*probably some time in the first half of 1845.

The first occurrence of

*vector*and

*scalar*in the

*London, Edinburgh, and Dublin Philosophical Magazine*is in volume XXIX (1846) in the article "On Quaternions; or on a new System of Imaginaries in Algebra":

The separation of the real and imaginary parts of a quaternion is an operation of such frequent occurrence, and may be regarded as being so fundamental in this theory, that it is convenient to introduce symbols which shall denote concisely the two separate results of this operation. The algebraicallyA prominent quaternionist, P. G. Tait, wrote, "Even Prof. Willard Gibbs must be ranked as one of the retarders of Quaternion progress..." After quoting this remark, M. J Crowe continues, "Gibbs did retard quaternion progress for his ...realpart may receive, according to the question in which it occurs, all values contained on the onescaleof progression from number from negative to positive infinity; we shall call it therefore thescalar part,or simply thescalarof the quaternion, and shall form its symbol by prefixing, to the symbol of the quaternion, the characteristic Scal., or simply S., where no confusion seems likely to arise from using this last abbreviation. On the other hand, the algebraicallyimaginarypart, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called thevector part,or simply thevectorof the quaternion; and may be denoted by prefixing the characteristic Vect. or V...<

/Q>

Information for this article was provided by David Wilkins and Julio González Cabillón. See HAMILTON and QUATERNION.

VECTOR ANALYSISappears in 1881 in J. W. Gibbs's (1839-1903)Elements of Vector Analysis Arranged for the Use of Students in Physics: "The numerical description of a vector requires three numbers, but nothing prevents us from using a single number for its symbolical designation. An algebra or analytical method in which a single letter or other expression is used to specify a vector may be called avector algebraorvector analysis."Scientific PapersII, (1906) p. 17.

Prefacing theElementsis the statement,

"The fundamental principles of the following analysis are such as are familiar under a slightly different form to students of quaternions. The manner in which the subject is followed is somewhat different ..., since the object of the writer does not require any use of the conception of the quaternion, being simply to give a suitable notation for those relations between vectors, or between vectors and scalars, which seem most important, and which lend themselves most readily to analytical transformations, and to explain some of these transformations."

*Elements of Vector Analysis*marks the beginning of modern vector analysis." (

*A History*

*of Vector Analysis*(2

^{nd}edition, p. 150)) Crowe considers Oliver Heaviside (1850-1925) an independent and nearly simultaneous creator of the modern system.

For

**vector analysis**entries on the

*Words*pages, see here for a list. See also the Earliest Uses of Symbols of Calculus

**VECTOR FIELD**is found in 1905 in "The Present Problems of Geometry" by Dr. Edward Kasner in

*Congress of Arts and Science, Universal Exposition, St. Louis, 1904*: "The vector field deserves to be introduced as a standard form into geometry." [Google print search]

**VECTOR PRODUCT**and

**SCALAR PRODUCT**are found in 1878 in William Kingdon Clifford's (1845-1879)

*Elements of*

*Dynamic*(1878, p. 95): "We are led to two different kinds of product of two vectors,..a vector product..and a scalar product." (OED2).

M. J Crowe

*A History of Vector Analysis*(2

^{nd}edition, pp. 139-43) finds an ambiguity in Clifford's treatment of the scalar product and argues that the modern definition only appears in the posthumously published

*Common Sense of the Exact Sciences*(1885), the relevant part of which was written by the editor, Karl Pearson. Crowe presents Clifford as a transitional figure in the movement from QUATERNION algebra to VECTOR ANALYSIS.

**VECTOR TRIPLE PRODUCT**and

**SCALAR TRIPLE PRODUCT**appear in 1901 in E. B. Wilson,

*Vector Analysis: A Text Book for the Use of Students of Mathematics and Physics Founded upon the Lectures of J. Willard Gibbs*: "The second triple product is the scalar product of two vectors, of which one is itself a vector product, as

**A**· (

**B**×

**C**) or (

**A**×

**B**) ·

**C**. This sort of product has a scalar value and consequently is often called the scalar triple product." (p. 68) and then on p. 71: "The third type of triple product is the vector product of two vectors of which one is itself a vector product. Such are

**A**× (

**B**×

**C**) and (

**A**×

**B**) ×

**C**.... This product is termed the vector triple product in contrast to the scalar product." Both products had appeared in Gibbs's

*Elements of Vector Analysis*but were not given names.

See VECTOR ANALYSIS and also OTHER SYMBOLS OF OPERATION on Earliest Uses of Symbols of Operation.

**VECTORS**and

**VECTOR SPACE.**At the beginning of the 20

^{th}century

*vector*usually meant a 3-dimensional object, though occasionally the term might be applied to objects in 2 or

*n*-dimensions. The situation is illustrated in the writing of Maxime Bôcher. In "The Theory of Linear Dependence" Bôcher wrote that "the simplest geometrical interpretation for a complex quantity with

*n*principal units is as a vector in space of

*n*dimensions." (

*Annals of Mathematics*,

**2**, 1900/1, p. 83.) yet there are no vectors in his

*Introduction to Higher Algebra*(1907), where

*n*-tuples of numbers are called "complex quantities" or "sets of numbers." It was years before the use of vector terminology became standard in discussions of

*n*-dimensional and abstract linear spaces.

The study of linear spaces began around the same time as Hamilton's first work on quaternions with Hermann Grassmann's

*Die Ausdehnungslehre*(1844). In 1888 G. Peano gave an abstract treatment in the 20

^{th}century manner in

*Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle operazioni della logica deduttiva*. See the St. Andrews page Hermann Weyl's

*Raum*,

*Zeit*,

*Materie*(

*Space, Time, Matter,*4

^{th}edition 1921) expressed Grassmann content using the language of vectors.

*Vector*and

*scalar product*both appear in the presentation of

*n*-dimensional "affine geometry," the systematic treatment of which had started with Grassmann's "bahnbrechende" book. Vector terminology also appears in Courant and Hilbert's

*Methoden der Mathematischen Physik*(1924, p. 1): "Ein System von

*n*reellen Zahlen

*x*

_{1}, ...,

*x*

_{n}nennen wir einer

*n-dimensionalen Vektor*oder einen Vector im Raume von

*n*Dimensionen ..." Courant and Hilbert used, however, Grassmann's term INNER PRODUCT.

Functional analysis and abstract algebra gave vector terminology extra life. Chapter II of Banach's

*Théorie des Operations Linéaires*(1932) is called "Espaces vectoriels généraux," although Banach did not call the elements of the spaces "vectors." B. L. van der Waerden

*Moderne Algebra I*(2

^{nd}edition, 1937) has a section on vector spaces. Paul R. Halmos's

*Finite Dimensional Vector Spaces*(1942,

*Annals of Mathematical Studies 7*) was inspired mainly by von Neumann's work on HILBERT SPACE.

[This entry was contributed by John Aldrich. For more information see two articles in

*Historia Mathematica*,

**22**, (1995): J.-L. Dorier "A General Outline of the Genesis of Vector Space Theory" (pp. 227-261) and G. H. Moore "The Axiomatization of Linear Algebra: 1875-1940" (pp. 262-305).]

**VENN DIAGRAM.**The diagrams now known as Venn diagrams were introduced in July 1880 in John Venn: "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings,"

*Philosophical Magazine and Journal of Science,*X (1880), 1-18.

In chapter V of his

*Symbolic Logic*(1881) John Venn showed a "reformed scheme of diagrammatic notation" based on what has become known as the Venn diagram. According to Venn, "the majority of modern logical treatises make at any rate occasional appeal to diagrammatic aid ..." but he considered the most popular scheme, the "Eulerian," inapplicable for "the purposes of a really general Logic."

OED 2 traces the evolution of the term

*Venn diagram*through the following quotations: "The application of Mr Venn's diagrammatic scheme to syllogistic reasonings" in J. N. Keynes

*Studies & Exercises in Formal Logic*III. v. 207 (1884); "Syllogisms in

*Barbara*and

*Camestres*may be taken in order to show how Dr Venn's diagrams can be used."

*Ibid.*(ed. 3) III. iv. 298 (1894); "This method resembles nothing so much as solution by means of the Venn diagrams." C. I. Lewis

*Survey Symbolic Logic*i. 77 (1918).

Information for this entry was contributed by John Aldrich and Wilfried Neumaier. See also

*Euler diagram.*

**VERSED SINE.**According to Smith (vol. 2, page 618), "This function, already occasionally mentioned in speaking of the sine, is first found in the

*Surya Siddhanta*(c. 400) and, immediately following that work, in the writings of Aryabhata, who computed a table of these functions. A sine was called the

*jya*; when it was turned through 90 degrees and was still limited by the arc, it

became the turned (versed) sine,

*utkramajya*or

*utramadjya.*"

Albategnius (al-Battani, c. 920) uses the expression "turned chord" (in some Latin translations

*chorda versa*).

The Arabs spoke of the

*sahem,*or arrow, and the word passed over into Latin as

*sagitta.*

Boyer (page 278) seems to imply that

*sinus versus*appears in 1145 in the Latin translation by Robert of Chester of al Khwarizmi's Algebra, although Boyer is unclear.

In

*Practica geomitrae,*Fibonacci used the term

*sinus versus arcus.*According to Smith (vol. 2), Fibonacci (1220) used

*sagitta.*

Fincke used the term

*sinus secundus*for the versed sine.

Regiomontanus (1436-1476) used

*sinus versus*for the versed sine in

*De triangulis omnimodis*(On triangles of all kinds; Nuremberg, 1533).

Maurolico (1558) used

*sinus versus major*(Smith vol. 2).

The OED shows a use in 1596 in English of "versed signe" by W. Burrough in

*Variation of Compasse.*

The term

**VERSIERA**was coined by Luigi Guido Grandi (1671-1742) (DSB). See

*witch of Agnesi.*

The term

**VERSOR**was introduced by William Rowan Hamilton (1805-1865). It appears in his

*Elements of Quaternions*ii. i. (1866) 133: "We shall now say that every Radial Quotient is a Versor. A Versor has thus, in general, a plane, an axis, and an angle." (

*OED*and Julio González Cabillón.) Unlike VECTORand SCALAR, two of Hamilton's other QUATERNION terms,

*versor*did not enter general currency.

**VERTEX**occurs in English in 1570 in John Dee's preface to Billingsley's translation of Euclid (OED2).

In 1828,

*Elements of Geometry and Trigonometry*(1832) by David Brewster (a translation of Legendre) has:

In ordinary language, the wordangleis often employed to designate the point situated at the vertex. This expression is inaccurate. It would be more correct and precise to use a particular name, such as that ofverticesfor designating the points at the corners of a polygon or of a polyedron. The denominationvertices of a polyedron,as employed by us, is to be understood in this sense.

**VERTICAL ANGLE**is found in English in 1571 in

*A Geometrical Practical Treatize Named Pantometria*edited by Thomas Digges: "Two right lines crossing one another, make the contrary or verticall angles equall" [OED2].

The term appears in English translations of Euclid.

**VIGINTIANGULAR.**The OED2 shows one citation, from 1822, for this term, meaning "having 20 angles." The word also appears in

*Webster's New International Dictionary,*2nd ed. (1934).

**VINCULUM**and

**VIRGULE.**In the Middle Ages, the horizontal bar placed over Roman numerals was called a

*titulus.*The term was used by Bernelinus. It was used more commonly to distinguish numerals from words, rather than to indicate multiplication by 1000.

Fibonacci used the Latin words

*virga*and

*virgula*for the horizontal fraction bar.

Tartaglia (1556) used

*virgoletta*for the horizontal fraction bar (Smith vol. 2, page 220).

In 1594 Blundevil in

*Exerc.*(1636) referred to the fraction bar as a "little line": "The Numerator is alwayes set above, and the Denominator beneath, having a little line drawne betwixt them thus $\large\frac{1}{2}\normalsize$ which signifieth one second or one halfe" (OED2).

In 1660 J. Moore in

*Arith.*used

*separatrix*for the line that was then placed after the units digit in decimals: "But the best and most distinct way of distinguishing them is by a rectangular line after the place of the unit, called Seperatrix. ... Therefore in writing of decimall parts let the seperatrix be always used" (OED2).

In 1696, Samuel Jeake referred to the fraction bar as "the intervening line" in his

*Arithmetick.*

In 1771

*separatrix*was used for the fraction bar in Luckombe,

*Hist. Printing*: "The Separatrix, or rule between the Numerator and Denominator [of fractions]" (OED2).

Leibniz, writing in Latin, used

*vinculum*for the grouping symbol.

In mathematics,

*vinculum*originally referred only to the grouping symbol, but some writers now use the word also to describe the horizontal fraction bar.

**VITAL STATISTICS.**The term entered currency in the 1830s. The

*OED*quotes William Farr writing in 1837 in M

^{c}Culloch's

*Descriptive and Statistical Account of the British Empire*II. 567: "Vital Statistics; or, the Statistics of Health, Sickness, Diseases, and Death." Vital statistics were one of the main interests of the statistical societies founded around that time: including the London Statistical Society and the American Statistical Association. See STATISTICS

**VON MISES DISTRIBUTION**and

**VON MISES-FISHER DISTRIBUTION.**These terms are used for directional distributions that are analogues of the normal distribution. The distribution on the circle was introduced by R. von Mises in 1918 in connection with the hypothesis that atomic weights are integers subject to error: "Über die 'Ganzahligkeit' der Atomgewicht und vervante Fragen,

*Physikal. Z*.,

**18**, 490-500.

The modern interest in these distributions dates from the early 1950s. E. J. Gumbel, J. A. Greenwood & D. Durand wrote about "The Circular Normal Distribution: Theory and Tables,"

*Journal of the American Statistical Association*,

**48**, (1953), 131-152. Greenwood & Durand refer to this as the "Mises distribution law" in "The Distribution of Length and Components of the Sum of

*n*Random Unit Vectors,"

*Annals of Mathematical Statistics*,

**26**, (1955), 233-246.

The distribution on the sphere was developed by R. A. Fisher in 1953 to treat problems in paleomagnetism: Dispersion on a Sphere,

*Proceedings of the Royal Society, A*,

**217**, 295-305. Fisher appears not to have known of von Mises's work or, indeed, of Langevin's derivation of the spherical distribution in 1905 for a problem in statistical mechanics.

The name von Mises-Fisher distribution was used by P. Hartman and G. S. Watson to refer to the

*n*-dimensional

*generalisation*of the von Mises and Fisher distributions: "'Normal' Distribution Functions on Spheres and the Modified Bessel Functions,"

*Annals of Probability*,

**2**, (1974), 593-607

[This entry was contributed by John Aldrich, based on K. V. Mardia "Directional Distributions"

*Encyclopedia of the Statistical Sciences*, vol 2 (1982).]

The term

**VON NEUMANN ALGEBRAS**was used by Jacques Dixmier in 1957 in

*Algebras of operators in Hilbert space (von Neumann algebras).*The term is named for John von Neumann, who had used the term "rings of operators." Another term is "W-algebras."

**VULGAR FRACTION.**In Latin, the term was

*fractiones vulgares,*and the term originally was used to distinguish an ordinary fraction from a sexagesimal.

Trenchant (1566) used

*fraction vulgaire*(Smith vol. 2, page 219).

Digges (1572) wrote "the vulgare or common Fractions."

Sylvester used the term in

*On the theory of vulgar fractions,*Amer. J. Math. 3 (1880).

The term

*common fraction*is now more widely used.