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

The term Banach space was coined by Maurice Fréchet (1878-1973), according to the University of St. Andrews website.

Banach space is found in National Mathematics Magazine in November 1945 in the title "Integrations of Functions in a Banach Space" by M. S. Macphail.

BANACH-TARSKI is found in Waclaw Sierpinski, "Sur le paradoxe de MM. Banach et Tarski," Fundamenta Mathematicae 33, pp 229-234 (1945) [James A. Landau].

BAR CHART occurs in Nov. 1914 in W. C. Brinton, "Graphic Methods for Presenting Data. IV. Time Charts," Engineering Magazine, 48, 229-241 (David, 1998).

The form of diagram, however, is much older; there is an example from William Playfair's Commercial and Political Atlas of 1786 at http://www.york.ac.uk/depts/maths/histstat/playfair.gif.

BAR GRAPH is dated 1924 in MWCD10.

Bar graph is found in 1925 in Statistics by B. F. Young: "Bar-graphs in the form of progress charts are used to represent a changing condition such as the output of a factory" (OED2).

The term BARYCENTRIC CALCULUS appears in 1827 in the title Der barycentrische calkul by August Ferdinand Möbius (1790-1868).

BASE (of a geometric figure) appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements (OED2).

BASE (in an isosceles triangle) is found in English in 1571 in Digges, Pantom.: "Isoscheles is such a Triangle as hath onely two sides like, the thirde being vnequall, and that is the Base" (OED2).

BASE (in logarithms) appears in Traité élémentaire de calcul différentiel et de calcul intégral (1797-1800) by Lacroix: "Et si a désigne la base du système, il en résulte l'équation y = a^x, dans laquelle les logarithmes sont les abscisses."

Base is found in the 1828 Webster dictionary, in the definition of radix: "2. In logarithms, the base of any system of logarithms, or that number whose logarithm is unity."

BASE (of a number system). Radix was used in the sense of a base of a number system in 1811 in An Elementary Investigation of the Theory of Numbers by Peter Barlow [James A. Landau].

Base is found in the Century Dictionary (1889-1897): "The base of a system of arithmetical notation is a number the multiples of whose powers are added together to express any number; thus, 10 is the base of the decimal system of arithmetic."

BASE ANGLE is found in 1848 in "On the Formation of the Central Spot of Newton's Rings Beyond the Critical Angle" by Sir George Gabriel Stokes in the Transactions of the Cambridge Philosophical Society [University of Michigan Historic Math Collection].

BASIS (of a vector space). The term basis-system was used by Frobenius and Stickelberger in 1878 in Crelle, according to Moore (1896) [James A. Landau].

BAYES ESTIMATE, BAYES SOLUTION in statistical decision theory. Wald ("Contributions to the Theory of Statistical Estimation and Testing  Hypotheses," Annals of Mathematical Statistics, 10, (1939), 299-326) originally used the term "minimum risk estimate" for what Hodges & Lehmann called a Bayes estimate ("Some Problems in Minimax Point Estimation," Annals of Mathematical Statistics, 21, (1950), 182-197.) Wald had used the term "Bayes solution" (in a more general setting) in his "An Essentially Complete Class of Admissible Decision Functions" Annals of Mathematical Statistics, 18, (1947), 549-555. Hodges & Lehmann (Annals of Mathematical Statistics, 19, 396-407) used the term BAYES RISK for a concept Wald had treated in 1939 without naming it [John Aldrich, based on David (2001)].

BAYES FACTOR appeared in I. J. Good’s 1958 "Significance tests in parallel and in series" Journal of the American Statistical Association, 53, 799-813. Previously in his Probability and the Weighing of Evidence (1950) Good had used the term "factor" explaining that "Dr. A. M. Turing suggested in a conversation in 1940 that the word 'factor' should be regarded as a technical term ... and that it could be more fully described as the factor in favour of the hypothesis H in virtue of the result of the experiment" [John Aldrich, using David 2001].

BAYES'S RULE is found in 1863 in An outline of the necessary laws of thought: a treatise on pure and applied logic by William Thomson: "The probability that there exist a cause of the reproduction of any event observed several times in succession is expressed by a fraction which has for its denominator the number 2 multiplied by itself as many times as the event has been observed, and for its numerator the same product minus one. This has been called Bayes's rule, and its validity is not so generally admitted as that of the preceding ones" [University of Michigan Historic Math Collection].

BAYES'S THEOREM. Règle de Bayes appears in 1843 in Exposition de la Théorie des Chances et des Probabilités by A. A. Cournot (David, 1998).

Bayes's Theorem appears in English in 1865 in A History of the Mathematical Theory of Probability by Isaac Todhunter (David, 1995).

BAYESIAN is found in 1950 in Contributions to Mathematical Statistics by R. A. Fisher. Fisher, a critic of the Bayesian approach, was distinguishing the probabilities used in the Bayesian argument from those generated by his own fiducial argument. The Bayesian argument used to be called the "method of inverse probability" or "inverse method." The language has changed and the reputation of Bayes (1702-61) has risen since Augustus De Morgan (An Essay on Probabilities (1838), p. vii) wrote "This [inverse] method was first used by the Rev. T. Bayes ... [who], though almost forgotten, deserves the most honourable remembrance from all who treat the history of this science" [John Aldrich, using David, 1998].

BELL-SHAPED. Bell-shaped parabola appears in 1857 in Mathematical Dictionary and Cyclopedia of Mathematical Science. The equation is ay² - x² + bx² = 0.

Bell-shaped parabola appears in an 1860 translation of a Latin work of Isaac Newton, Sir Isaac Newton's Enumeration of lines of the third order, generation of curves by shadows, organic description of curves, and construction of equations by curves [University of Michigan Historic Math Collection].

Bell-shaped curve is found in 1876 in Catalogue of the Special Loan Collection of Scientific Apparatus at the South Kensington Museum by Francis Galton (David, 1998).

J. V. Uspensky, in Introduction to Mathematical Probability (1937), writes that "the probability curve has a bell-shaped form" [James A. Landau].

BELL CURVE is dated ca. 1941 in MWCD10.

BERNOULLI NUMBERS. According to Cajori (vol. 2, page 42), Leonhard Euler introduced the name "Bernoullian numbers."

The term appears in 1769 in the title "De summis serierum numeros Bernoullianos involventium" by Leonhard Euler.

According to the University of St. Andrews website, in its article on Johann Faulhaber, the Bernoulli numbers were "so named by [Abraham] de Moivre" (1667-1754).

BERNOULLI TRIAL is dated 1951 in MWCD10, although James A. Landau has found the phrases "Bernoullian trials" and "Bernoullian series of trials" in 1937 in Introduction to Mathematical Probability by J. V. Upensky.

BESSEL FUNCTION. Franceschetti (p. 56) implies that this term was introduced by Oskar Xavier Schlömilch in 1854.

Bessel'schen Functionen appears in 1868 in the title Studien über die Bessel'schen Functionen by Eugen Lommel.

Philosophical Magazine in 1872 has "The value of Bessel's functions is becoming generally recognized" (OED2).

Bessel function appears in 1894 in Ann. Math. IX. 27 in the heading "Roots of the Second Bessel Function" (OED2).

BETA DISTRIBUTION. Distribuzione [beta] is found in 1911 in C. Gini, "Considerazioni Sulle Probabilità Posteriori e Applicazioni al Rapporto dei Sessi Nelle Nascite Umane," Studi Economico-Giuridici della Università de Cagliari, Anno III, 5-41 (David, 1998).

The term BETTI NUMBER was coined by Henri Poincaré (1854-1912) and named for Enrico Betti (1823-1892), according to a history note by Victor Katz in A First Course in Abstract Algebra by John B. Fraleigh.

BETWEENNESS. The earliest citation in the OED2 for this word is in 1892 Monist II. 243: "In reality there are not two things and, in addition to them a betweenness of the two things."

Betweenness appears in G. B. Halsted, "The betweenness assumptions," Amer. Math. Monthly 9, 98-101.

The OED2 has a 1904 citation which makes reference to "Hilbert's betweenness assumptions."

BEZOUTIANT was "used by Sylvester and later writers" (Cajori 1919, page 249).

BIASED and UNBIASED. Biased errors and unbiased errors (meaning "errors with zero expectation") are found in 1897 in A. L. Bowley, "Relations Between the Accuracy of an Average and That of Its Constituent Parts," Journal of the Royal Statistical Society, 60, 855-866 (David, 1995).

Biased sample is found in 1911 An Introduction to the theory of Statistics by G. U. Yule: "Any sample, taken in the way supposed, is likely to be definitely biassed, in the sense that it will not tend to include, even in the long run, equal proportions of the A’s and [alpha]'s in the original material" (OED2).

Biased sampling is found in F. Yates, "Some examples of biassed sampling," Ann. Eugen. 6 (1935) [James A. Landau].

The term BICURSAL was introduced by Cayley (Kline, page 938).

In 1873 Cayley wrote, "A curve of deficiency 1 may be termed bicursal."

BIJECTION and BIJECTIVE are dated 1966 in MWCD10.

BILLION. See million.

BIMODAL appears in 1903 in S. R. Williams, "Variation in Lithobius Forficatus," American Naturalist, 37, 299-312 (David, 1998).

BINARY ARITHMETIC appears in English in 1796 A Mathematical and Philosophical Dictionary (OED2).

BINOMIAL. According to the OED2, the Latin word binomius was in use in algebra in the 16th century.

Binomial first appears as a noun in English in its modern mathematical sense in 1557 in The Whetstone of Witte by Robert Recorde: "The nombers that be compound with + be called Bimedialles... If their partes be of 2 denominations, then thei named Binomialles properly. Howbeit many vse to call Binomialles all compounde nombers that have +" (OED2).

BINOMIAL COEFFICIENT. According to Kline (page 272), this term was introduced by Michael Stifel (1487-1567) about 1544. However, Julio González Cabillón believes this information is incorrect. He says Stifel could not have used the word coefficient, which is due to Vieta (1540-1603).

Binomial coefficient is found in Rottock, "Ueber Reihen mit Binomialcoefficienten und Potenzen," Pr. d. G. Rendsburg (1868).

Binomial coefficient is found in English in an 1868 paper by Arthur Cayley [University of Michigan Historical Math Collection].

BINOMIAL DISTRIBUTION is found in 1911 in An Introduction to the Theory of Statistics by G. U. Yule: "The binomial distribution,..only becomes approximately normal when n is large, and this limitation must be remembered in applying the table..to cases in which the distribution is strictly binomial" (OED2).

BINOMIAL THEOREM appears in 1742 in Treatise of Fluxions by Colin Maclaurin (Struik, page 339).

In Gilbert and Sullivan's The Pirates of Penzance (1879), the song "I Am The Very Model of a Modern Major-General" includes the lines:

I'm very well acquainted, too, with matters mathematical,
I understand equations, both the simple and quadratical,
About binomial theorem I'm teeming with a lot o' news,
With many cheerful facts about the square of the hypotenuse. [...]
I'm very good at integral and differential calculus;
I know the scientific names of beings animalculous:

The term BIOMATHEMATICS was coined by William Moses Feldman (1880-1939), according to Garry J. Tee in "William Moses Feldman: Historian of Rabbinical Mathematics and Astronomy." The term appears in Feldman's textbook Biomathematics published in 1923.

BIOSTATISTICS appears in Webster's New International Dictionary (1909).

BIPARTITE. In 1858, Cayley referred to "bipartite binary quantics."

BIPARTITE CURVE appears in 1879 in George Salmon (1819-1904), Higher Plane Curves (ed. 3): "We shall then call the curve we have been considering a bipartite curve, as consisting of two distinct continuous series of points" (OED2).

BIQUATERNION. Hamilton used the term biquaternion in the sense of a quaternion with complex coefficients.

In the more recent sense, William Kingdon Clifford (1845-1879) coined the term. It appears in 1873 in Proc. London Math. Soc. IV. 386.

BISECT. According to the OED2, bisect is apparently of English formation. The word is dated ca. 1645 in MWCD10.

Bisection appears in 1656 in a translation of Hobbes's Elem. Philos. (1839) 307: "By perpetual bisection of an angle" (OED2).

In 1660, Barrow's translation of Euclid's Elements has "To bisect a right line."

Bisector appears in English in 1864 in The Reader 5 Oct. 483/2: "The internal and external bisectors of the angle" (OED2).

BIT was coined by John W. Tukey (1915-2000).

According to Niels Ole Finnemann in Thought, Sign and Machine, Chapter 6, "After some more informal contacts during the first war years, on the initiative of mathematician Norbert Wiener, a number of scientists gathered in the winter of 1943-44 at a seminar, where Wiener himself tried out his ideas for describing intentional systems as based on feedback mechanisms. On the same occasion J. W. Tukey introduced the term a 'bit' (binary digit) for the smallest informational unit, corresponding to the idea of a quantity of information as a quantity of yes-or-no answers."

Several Internet web pages say Tukey coined the term in 1946. Another web page says, "Tukey records that it evolved over a lunch table as a handier alternative to 'bigit' or 'binit.'"

Bit first appeared in print in July 1948 in "The Mathematical Theory of Communication" by Claude Elwood Shannon (1916-2001) in the Bell Systems Technical Journal. In the article, Shannon credited Tukey with the coinage [West Addison assisted with this entry.]

BIVARIATE is found in 1920 in Biometrika XIII. 37: "Thus in 1885 Galton had completed the theory of bi-variate normal correlation" (OED2).

BOOLEAN is found in 1851 in the Cambridge and Dublin Mathematical Journal vi. 192: "...the Hessian, or as it ought to be termed, the first Boolian Determinant" (OED2).

BOOLEAN ALGEBRA. Boolian algebra appears in the Century Dictionary (1889-1897):

Boolian algebra, a logical algebra, invented by the English mathematician George Boole (1815-64), for the solution of problems in ordinary logic. It has also a connection with the theory of probabilities.

In an illuminating passage of "Algebraic Logic", Halmos writes (p. 11):

Terminological purists sometimes object to the Boolean use of the word "algebra". The objection is not really cogent. In the first place, the theory of Boolean algebras has not yet collided, and it is not likely to collide, with the theory of linear algebras. In the second place, a collision would not be catastrophic; a Boolean algebra is, after all, a linear algebra over the field of integers modulo 2. (...) While, to be sure, a shorter and more suggestive term than "Boolean algebra" might be desirable, the nomenclature is so thoroughly established that to change now would do more harm than good.

BOOTSTRAP in Statistics. The term was introduced by Bradley Efron in "Bootstrap methods: another look at the jackknife," Annals of Statistics, 7, (1979) 1-26. Tukey’s "jackknife" had set a precedent for "colorful" terminology and Efron reported some suggestions for his construct: "Swiss Army Knife, Meat Axe, Swan-Dive, Jack Rabbit and my personal favorite, the Shotgun, which to paraphrase Tukey, 'can blow the head off any problem if the statistician can stand the resulting mess.'" In An Introduction to the Bootstrap (with R. J. Tibshirani) (1993) Efron explained that "the use of the term bootstrap derives from the phrase to pull oneself up by one's own bootstrap, widely thought to be based on one of the eighteenth century Adventures of Baron Munchausen, by Rudolph Erich Raspe. (The Baron had fallen to the bottom of a deep lake. Just when it looked like all was lost, he thought to pick himself up by his own bootstraps.)" [John Aldrich]

BORROW is found in English in 1594 in Blundevil, Exerc.: "Take 6 out of nothing, which will not bee, wherefore you must borrow 60" (OED2).

In October 1947, "Provision for Individual Differences in High School Mathematics Courses" by William Lee in The Mathematics Teacher has: "The Social Mathematics course stresses understanding of arithmetic: 'carrying' in addition, 'regrouping' (not 'borrowing') in subtraction, 'indenting' in multiplication are analyzed and understood rather than remaining mere rote operations to be performed blindly."

BOYER'S LAW appears in H. C. Kennedy, "Boyer's Law: Mathematical formulas and theorems are usually not named after their original discoverers," Amer. Math. Monthly, 79:1 (1972), 66-67.

Boyer's theorem is found in 1968 in History of Mathematics by Barnabas Hughes.

The term BRACHISTOCHRONE was introduced by Johann Bernoulli (1667-1748). Smith (vol. 2, page 326) says the term is "due to the Bernoullis."

The terms BRA VECTOR and KET VECTOR were introduced by Paul Adrien Maurice Dirac (1902-1984). The terms appear in 1947 in Princ. Quantum Mech. by Dirac: "It is desirable to have a special name for describing the vectors which are connected with the states of a system in quantum mechanics, whether they are in a space of a finite or an infinite number of dimensions. We shall call them ket vectors, or simply kets, and denote a general one of them by a special symbol >|. ... We shall call the new vectors bra vectors, or simply bras, and denote a general one of them by the symbol <|, the mirror image of the symbol for a ket vector" (OED2).

BRIGGSIAN LOGARITHM. The phrase Briggs logarithm is found in the 1771 edition of the Encyclopaedia Britannica [James A. Landau].

BROKEN LINE is found in 1852 in Elements of the differential and integral calculus by Charles Davies: "But the arc POM can never be less than the chord PM, nor greater than the broken line PNM which contains it; hence, the limit of the ratio POM/PM = 1; and consequently, the differential of the arc is equal to the differential of the chord."

Broken line is found in 1852 in Legendre, A. M. (Adrien Marie): Elements of geometry and trigonometry, from the works of A. M. Legendre. Revised and adapted to the course of mathematical instruction in the United States, by Charles Davies" "5. A Straight Line is one which lies in the same direction between any two of its points. 6. A Broken Line is one made up of straight lines, not lying in the same direction."

Broken line is found in 1852 in Elements of plane trigonometry, with its application to mensuration of heights and distances, surveying and navigation by William Smyth: "Instead of a broken line, a field is sometimes bounded by a line irregularly curves, as by the margin of a brook, river, or lake. In this case (fig. 60) we run, as before, a chain line as near the boundary as possible, and by means of offsets determine a sufficient number of points in the curve to draw it." [These three citations were found using the University of Michigan Historic Math Collection.]

According to Schwartzman (page 38), the "broken line," meaning a curve composed of connected straight line segments, was adopted "around 1898" by David Hilbert (1862-1943).

BROWNIAN MOTION. In the course of the 20th century the physical phenomenon described by Brown in 1827 was described in mathematical terms and gradually "Brownian motion" came to refer as much to the mathematical formalism as to the phenomenon. Mathematical theories were developed by, inter alia, A. Einstein ("Zur Theorie der Brownschen Bewegung" (1905)). The "Brownian motion process" of J. L. Doob's Stochastic Processes (1954) is a type of stochastic process divested of physical application. Doob states that the process "was first discussed by Bachelier and later, more rigorously by Wiener. It is sometimes called the Wiener process." An earlier term in physics (and mathematics) was "Brownian movement." This slowly gave way to "Brownian motion," although David (2001) reports an early appearance of "Brownian motion" in 1892 in W. Ramsay's Report of a paper read to the Chemical Society, London. Nature, 45, 429/2. (See Wiener process.) [John Aldrich]

BRUN'S CONSTANT was coined by R. P. Brent in "Irregularities in the distribution of primes and twin primes," Math. Comp. 29 (1975), according to Algorithmic Number Theory by Bach and Shallit [Paul Pollack].

The term BYTE was coined in 1956 by Dr. Werner Buchholz of IBM. A question-and-answer session at an ACM conference on the history of programming languages included this exchange:

JOHN GOODENOUGH: You mentioned that the term "byte" is used in JOVIAL. Where did the term come from?
JULES SCHWARTZ (inventor of JOVIAL): As I recall, the AN/FSQ-31, a totally different computer than the 709, was byte oriented. I don't recall for sure, but I'm reasonably certain the description of that computer included the word "byte," and we used it.
FRED BROOKS: May I speak to that? Werner Buchholz coined the word as part of the definition of STRETCH, and the AN/FSQ-31 picked it up from STRETCH, but Werner is very definitely the author of that word.
SCHWARTZ: That's right. Thank you.