早期數學字彙的歷史 (V)

Last revision: June 30, 1999

VANDERMONDE DETERMINANT. According to the University of St. Andrews website, this term was introduced by Henri L廩n Lebesgue (1875-1941).

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 M嶰hanique 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 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.
The following is from page 8:
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 to denote the distance of an astronomical object from the sun or the earth. Hamilton makes it clear that he is introducing the term vector with a new sense, involving both length and direction. He explains this in, for example, section 16 of Lecture I of his "Lectures on Quaternions." He expands on this distinction in article 17, which includes the following:
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.

Hamilton, in his Lectures on Quaternions, is not satisfied with having introduced vector. Within a few pages we find vectum, vehend, revector, provector, provectum, transvehend, transvectum, etc., and identities such as
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 [...]
Next Hamilton goes on to tell us about another "chief class" of the "geometrical quotients," namely
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 [...]
David Wilkins believes that the paper "On quaternions" in the Proceedings of the Royal Irish Academy probably appeared earlier than the 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):

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...
(Information for this article was provided by David Wilkins and Julio Gonz嫮ez Cabill鏮.)

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 Gnter 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 Gonz嫮ez Cabill鏮.)

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.

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