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

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 appears in 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. Edgeworth used fluctuation for the square of the standard deviation.

Variance was introduced by Ronald Aylmer Fisher in 1918 in "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."

VARIATE appears in 1909 in Karl Pearson, "On a New Method of Determining Correlation...," Biometrika, 7, 96-105 (David, 1998).

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].

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).

Radius vector appears in a mathematical sense in 1831 in Elements of the Integral Calculus (1839) by J. R. Young: "...when the angle [omega] between the radius vector and fixed axis is taken for the independent variable, the formula is...."

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 having x, y, z for 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."

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...

But 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.

Provectum = Provector + Vector + Vehend.

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 of scalars to 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 a vector, 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 real part may receive, according to the question in which it occurs, all values contained on the one scale of progression from number from negative to positive infinity; we shall call it therefore the scalar part, or simply the scalar of 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 algebraically imaginary part, 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 the vector part, or simply the vector of the quaternion; and may be denoted by prefixing the characteristic Vect. or V...

VECTOR ANALYSIS occurs in 1881 in the title Elements of Vector Analysis by J. W. Gibbs.

The OED2 shows an 1881 quotation from J. W. Gibbs Scientific Papers (1906): "An algebra or analytical method in which a single letter or other expression is used to specify a vector may be called a vector algebra or vector analysis."

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. Euler's scheme of notation is found in 1858 in Elements of logic by Henry Coppée (1821-1895): "Euler's scheme of notation is altogether the one best suited to our purpose, and we shall limit ourselves to the explanation of that. It is essentially an arrangement of three circles, to represent the three terms of a syllogism, and, by their combination, the three propositions" [University of Michigan Digital Library].

Euler's system of notation appears in 1863 in An outline of the necessary laws of thought: a treatise on pure and applied logic by William Thomson (University of Michigan Digital Library).

Euler's notation appears in about 1869 in The principles of logic, for high schools and colleges by Aaron Schuyler (University of Michigan Digital Library).

Euler's diagram appears in 1884 in Elementary Lessons in Logic by W. Stanley Jevons: "Euler's diagram for this proposition may be constructed in the same manner as for the proposition I as follows:..."

Euler's circles appears in 1893 in Logic by William Minto (1845-1893): "The relations between the terms in the four forms are represented by simple diagrams known as Euler's circles."

Euler's circles appears in October 1937 in George W. Hartmann, "Gestalt Psychology and Mathematical Insight," The Mathematics Teacher: "But in the case of 'Euler's circles' as used in elementary demonstrations of formal logic, one literally 'sees' how intimately syllogistic proof is linked to direct sensory perception of the basic pattern. It seems that the famous Swiss mathematician of the eighteenth century was once a tutor by correspondence to a dull-witted Russian princess and devised this method of convincing her of the reality and necessity of certain relations established deductively."

Venn diagram appears in 1918 in A Survey of Symbolic Logic by Clarence Irving Lewis: "This method resembles nothing so much as solution by means of the Venn diagrams" (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.)

Versor appears about 1865 in Sir W. R. Hamilton, Elem. 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" (OED2).

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 word angle is 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 of vertices for designating the points at the corners of a polygon or of a polyedron. The denomination vertices of a polyedron, as employed by us, is to be understood in this sense.

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 1/2 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.

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.