However, the DSB states that Lebesgue believed that the attribution of this determinant to Vandermonde was due to a misreading of his notation, implying Lebesgue did not introduce the term.

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

The term **VARIANCE** was introduced by Ronald Aylmer Fisher in 1918
in the *Transactions of the Royal Society of Edinburgh:* "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."

The word **VECTOR (in astronomy)** usually occurs as part of the
term *radius vector.*

*Vector* appears in English in a 1704 dictionary; *radius vector*
appears in English in a 1753 dictionary.

Laplace used *rayon vecteur* in his *Méchanique Celeste* (1799-1825).

William Rowan Hamilton used *radius vector* in article 14 of "On
a General Method in Dynamics; by which the Study of the Motions of all
free Systems of attracting or repelling Points is reduced to the Search
and Differentiation of one central Relation, or characteristic Function."
This paper was published in the Philosophical Transactions of the Royal
Society of London in 1834.

**VECTOR (in mathematics).** Both the terms *vector* and *scalar*
were introduced by William Rowan Hamilton (1805-1865).

Both terms appear in 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,

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* 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 [...]

The first occurrence of *vector* and *scalar* in the *London,
Edinburgh, and Dublin Philosophical Magazine* is in volume XXIX (1846):

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 algebraically(Information for this article was provided by David Wilkins and Julio González Cabillón.)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 araise 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...

**VECTOR FIELD** is found in "Natural Families of Trajectories: Conservative
Fields of Force," Edward Kasner, *Transactions of the American Mathematical
Society,* Vol. 10, No. 2. (Apr., 1909).

**VECTOR PRODUCT** and **SCALAR PRODUCT** are found in 1878 in
*Dynamic*
by William Kingdon Clifford (1845-1879) (OED2).

**VECTOR SPACE.** The notion of a vector space is due to Hermann
Günter Grassmann (1844).

Peano's *Geometrical Calculus* (1888) defines the notion and presumably
uses the term.

*Vector space* occurs in English in "On the Geometry of Planes
in a Parabolic Space of Four Dimensions," Irving Stringham, *Transactions
of the American Mathematical Society,* Vol. 2, No. 2. (Apr., 1901).

**VECTOR TRIPLE PRODUCT** occurs in 1901 in Gibbs and Wilson, *Vector
Analysis* (OED2).

**VENN DIAGRAM** appears in 1918 in *A Survey of Symbolic Logic*
by Clarence Irving Lewis (OED2).

**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 Khowarizmi'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)
(Julio González Cabillón.)

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

**VINCULUM.** 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 word *virga* for the horizontal fraction
bar.

Tartaglia (1556) used *virgoletta* for the horizontal fraction
bar (Smith vol. 2, page 220).

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.

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 (1903-1957), 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.