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

Last revision: May 16, 2019

RADIAN. According to Cajori (1919, page 484):

An isolated matter of interest is the origin of the term “radians”, used with trigonometric functions. It first appeared in print on June 5, 1873, in examination questions set by James Thomson at Queen’s College, Belfast. James Thomson was a brother of Lord Kelvin. He used the term as early as 1871, while in 1869 Thomas Muir, then of St. Andrew’s University, hesitated between “rads,” “radials” and “radians.” In 1874, T. Muir adopted “radians” after a consultation with James Thomson.

In a footnote, Cajori gives a reference to Nature, Vol. 83, pp. 156, 217, 459, 460 [Julio González Cabillón].

In a letter appearing in the April 7, 1910, Nature, Thomas Muir wrote: I wrote to him [i.e., to Alexander J. Ellis, in 1874], and he agreed at once for the form “radians,” on the ground that it could be viewed as a contraction for “radial angles.”

In a letter appearing in the June 16, 1910, Nature, James Thomson wrote: "I shall be very pleased to send Dr. Muir a copy of my father’s examination questions of June, 1873, containing the word “radians.” ...It thus appears that “radians” was thought of independently by Dr. Muir and my father, and, what is really more important than the exact form of the name, they both independently thought of the necessity of giving a name to the unit-angle" [Dave Cohen].

According to W. N. Roseveare, “The radian is an uninteresting angle—Lord Kelvin introduced the word merely as a convenience in lecturing, to avoid the long phrase ‘angle whose circular measure is.’” This quotation appears on page 133 of W. N. Roseveare, “On ‘circular measure’ and the product forms of the sine and cosine,” Mathematical Gazette 3 #49 (January 1905), 129-137. [Dave L. Renfro]

A post on the Internet indicated that Thomas Muir (1844-1934) claimed to have coined the term in 1869, and that Muir and Ellis proposed the term as a contraction of "radial angle" in 1874. A reference given was: Michael Cooper, "Who named the radian?", Mathematical gazette 76, no. 475 (1992) 100-101. I have not seen this article.

A 1991 Prentice-Hall high school textbook, Algebra 2, by Bettye C. Hall and Mona Fabricant has: "James Muir, a mathematician, and James T. Thomson, a physicist, were working independently during the late nineteenth century to develop a new unit of angle measurement. They met and agreed on the name radian, a shortened form of the phrase radial angle. Different names were used for the new unit until about 1900. Today the term radian is in common usage."

In 1876-79, the Globe encyclopaedia of universal information has, in the Circle article: "The unit, called a radian by Professor James Thomson, is that angle whose subtending arc is equal in length to the radius" [University of Michigan Digital Library].

The expressions RADICAL LINE ("Axe Radical"), RADICAL CENTER OF CIRCLES ("Centre radical des cercles"), and other related terms were coined (in French) by Louis Gaultier (Julio González Cabillón).

RADICAL. The word radical was used in English before 1668 by Recorde and others to refer to an irrational number.

RADICAL SIGN appears in English in 1669 in An Introduction to Algebra edited in 1668 by John Pell (1611-1685):

In the quotient subjoyn the surd part with its first radical Sign.

RADICAND is found in 1889 in George Chrystal, Algebra (ed. 2) I. x. 182: "We shall restrict the radicand, k, to be positive" (OED On Line).

Radicand also appears in an 1890 Funk and Wagnalls dictionary.

RADIOGRAM was one of several terms launched by Karl Pearson in his lectures of 1892. According to Stigler (1986, p. 327), “The syllabi show an abundance of fancy terminology: stigmograms, euthygrams, epipedograms, hormograms, topograms, stereograms, radiograms, and isodemotic lines.” Pearson soon lost interest in these diagrams, or in all but one of them—the HISTOGRAM. Some of the words have been re-invented, thus in Britain stereogram and radiogram became names for types of audio equipment! [James A. Landau].

RADIUS. In modern English radius is used both for the line joining the center of a circle to any point on its circumference and for the length of that line. The Greeks did not have a special term for radius in either sense; contrast DIAMETER. In Euclid Book 1, postulate 3 the term radius was not used as it is in this translation. The term for “distance” was used, that being thought sufficient (Heath vol. 1 p. 199 and Smith vol. 2, page 278). Archimedes called the radius "ek tou kentrou" (the [line] from the center) [Samuel S. Kutler].

The term “semidiameter” appears in Latin in the Ars Geometriae of Boethius (c. 510), according to Smith (vol. 2, page 278). The English word appears in 1551 in Pathway to Knowledge by Robert Recorde: “Defin., Diameters, whose halfe, I meane from the center to the circumference any waie, is called the semidiameter, or halfe diameter” (OED).

The word radius is a Latin word originally meaning a staff, rod or stake and, by extension, a ray or beam of light. (OED and Schwartzman). The word was used by Peter Ramus (1515-1572) in his 1569 publication of P. Rami Scholarium mathematicarum kibri unus et triginti, writing “Radius est recta a centro ad perimetrum” (Smith vol. 2, page 278; DSB; Johnson, page 158).

The OED‘s earliest reference to radius as a mathematical term in English is Hobbes writing in 1656, “Is the radius that describes the inner circles equal to the radius that describes the exterior?” Six lessons to the professors of mathematicks of the institution of Sir H. Savile, in the University of Oxford, Works. 1845 VII. 256.

[This entry was contributed by John Aldrich.]

RADIUS OF CONVERGENCE is found in English in 1891 in a translation by George Lambert Cathcart of the German An introduction to the study of the elements of the differential and integral calculus by Axel Harnack [University of Michigan Historical Math Collecdtion].

Interval of convergence is found in 1891 in An Introduction to the Study of the Elements of the Differential and Integral Calculus by Axel Harnack. [Google print search]

RADIUS OF CURVATURE. In his Introductio in analysin infinitorum (1748), Euler works with the radius of curvature and says that this is commonly called "radius of osculation" but also sometimes "radius of curvature." William C. Waterhouse provided this citation and points out that the idea and term were in use earlier.

Thomas Simpson (1710-1761) wrote, "An equation between the radius of curvature . . . and the angle it makes with a given direction, implies all the conditions of the form of the curve, though not of its position."

Radius of curvature appears in 1753 in Chambers Cyclopedia Supplement: "Curvature, This circle is called the circle of curvature..and its semidiameter, the ray or radius of curvature" [OED].

The term radius of curvature may have been used earlier by Christiaan Huygens and Isaac Newton, who wrote on the subject.

RADIX, ROOT, UNKNOWN, SQUARE ROOT. Late Latin writers used res for the unknown. This was translated as cosa in Italian, and the early Italian writers called algebra the Regola de la Cosa, whence the German Die Coss and the English cossike arte (Smith vol. 2, page 392).

Other Latin terms used in the Middle Ages for the uknown quantity and its square were radix, res, and census.

The term root was used by al-Khowarizmi; the word is rendered radix in Robert of Chester’s Latin translation of the algebra of al-Khowarizmi. Radix also is used in translations from Arabic to Latin by John of Seville, Gerard of Cremona, and Leonardo of Pisa. For an early English use of root, see addition.

Root (meaning "square root" or "cubic root" etc.) is found in English in 1557 in The whetstone of witte by Robert Recorde: "Thei onely haue rootes, whiche bee made by many multiplications of some one number by it self" [OED].

Square root is found in English in 1557 in The whetstone of witte by Robert Recorde: "The roote of a square nombere, is called a Square roote" [OED].

Radix, meaning "root," appears in English in 1571 in A geometrical practise, named Pantometria by Leonard Digges: "The Radix Quadrate of the Product, is the Hypothenusa" [OED].

Unknown was used by Fermat. In "Novus Secundarum et Ulterioris Ordinis Radicum in Analyticis Usus," Fermat wrote (in translation):

There are certain problems which involve only one unknown, and which can be called determinate, to distinguish them from the problems of loci. There are certain others which involve two unknowns and which can never be reduced to a single one; these are the problems of loci. In the first problems we seek a unique point, in the latter a curve. But if the proposed problem involves three unknowns, one has to find, to satisfy the question, not only a point or a curve, but an entire surface.

Unknown is found in English in 1676 in Glanvill, Ess.: "The degree of Composition in the unknown Quantity of the Æquation" [OED].

In Miscellanea Berolinensia (1710) Leibniz used the phrase "incognita, x,."

Root (meaning "unknown") is found in English in 1728 in Chambers Cyclopedia: "The Root of an Equation, is the Value of the unknown Quantity in the Equation."

In the 1939 movie The Wizard of Oz, the Scarecrow says, “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.”

RADIX (base in logarithms). Elements of Navigationby John Robertson (1754) has: "The number usually set down, as Napier’s logarithm of 10, is more properly his logarithm divided by the radius of his trigonometrical table, which is the radix from which he raised his logarithms." [Alan Hughes, associate editor of the OED]

RADON-NIKODYM THEOREM in measure theory is named for Johann Radon and Otton Nikodym. R. M. Dudley Real Analysis and Probability (2002, p. 184) writes that, “The Radon-Nikodym theorem is due to Radon (1913, pp. 1342-1351), Daniell (1920) and Nikodym (1930).” The references are Radon “Theorie und Anwendungen der absolut additiven Mengenfunktionen,” Akad. Wiss. Sitzungsber. Kaiserl.Math.-Nat. Kl. 122 1295-1438, reprinted in his Gesammelte Abhandlungen, 1, 45-188. Birkhäuser, Basel, 1987; Daniell “Stieltjes Derivatives,” Bull. Amer. Math. Soc. 26, 444-448; Nikodym “Sur une généralisation des intégrales de M. J. Radon,” Fundamenta Mathematicae, 15, (1930), 131-179.

A JSTOR search found the term “Radon-Nikodym theorem” in use in 1939.

RANDOM DISTRIBUTION is found in 1854 in An Investigation of the Laws of Thought by George Boole: "If they have not, we may regard the phaenomenon of a double star as the accidental result of a ’srandom distribution’s of stars over the celestial vault, i.e. of a distribution which would render it just as probable that either member of the binary system should appear in one spot as in another." [Google print search]

RANDOM NUMBER. On July 10, 1897, the Fort Wayne News has: “‘1310,’ you answer, choosing a random number.”

Random number appears in the first book of random numbers, by Tippett: "In order to form this table of random numbers 40,000 digits were taken at random from census reports and combined by fours to give 10,000 numbers." p. iii of L. H. C. Tippett’s "Random Sampling Numbers," Tracts for Computers, No. 15 (1927). The work was done in Karl Pearson’s Biometric Laboratory where previously random numbers used in "experimental sampling" were generated by physically drawing marked tickets from a bag or bowl. [OED2, James A. Landau].

Random number now usually refers to what was originally called a pseudo-random number. The OED’s earliest quotation for pseudo-random number is from 1949, Seminar on Sci. Computation, Nov. (Internat. Business Machines) 104/2 "A random number c lying between 0 and 1 is selected from a store, or a pseudo~random number c lying between 0 and 1 is computed arithmetically."

See the entry MONTE CARLO.

RANDOM PROCESS. A Google search finds this phrase in 1802 in Observations and reflections on Storms, and some other Phenomena of the Atmosphere. Letter from Professor Waterhouse to Dr. Mitchell. The Universal Magazine of Knowledge and Pleasure, Volume 111 (1802) p 290-292. “They seem also to have known that mountains made a part of this grand apparatus, and to have believed that it was not a fortuitous or random process, but regulated, as we now kind it, by weight and measure.” [Herbert Acree]

Random process is found in Harald Cramér, "Random variables and probability distributions," Cambridge Tracts in Math. and Math. Phys. 36 (1937).

RANDOM SAMPLE. Random sample is found in 1847 in Introduction to the Literature of Europe in the Fifteenth, Sixteenth, and Seventeenth Centuries by Henry Hallam: "This is a random sample of Feltham’s style..." [Google print search]

Random sample and random selection are found in “A theory of probable Inference” by C. S. Pierce which appears in the 1883 book Studies in Logic by Members of the Johns Hopkins University and edited by Charles Sanders Peirce.

In such as case, it is our interest that fulfils the function of an apparatus for random selection; and no better need be desired, so long as we have reason to deem the premise “the proportion ρ of the M’s are P’s” to be equally true in regard to that part of the M’s which are alone likely ever to excite our interest. (Page 130)

In point of fact P’, P”, P’’’, etc. constitute a random sample of the characters of M, and the ratio r of them being found to belong to S, the same ratio of all the characters of M are concluded to belong to S. (Page 141)

[Google print search, Herbert Acree]

Random sampling is found in a review of “Studies in Logic” in Science, Vol 1, No. 18 (1883), p 514-516.

When we conclude that continuous random sampling of a given natural class must leads us towards discovering the true proportion of cases of the presence of a predestinated character in individuals of the class; must we not base our conclusion on the ultimate a priori assumption that our instinctive tendencies to observe natural facts are such as, in the longrun, will lead us to actual choice at random, and not to a choice unconsciously vitiated by unknown preference for cases that favor the conclusion we reach?

[Google print search, Herbert Acree]

Random selection occurs in 1897 in Karl Pearson & L. N. G. Filon Mathematical Contributions to the Theory of Evolution. IV. On the Probable Errors of Frequency Constants and on the Influence of Random Selection on Variation and Correlation. [Abstract] Proceedings of the Royal Society of London, Vol. 62. (1897 - 1898), p. 176: "A random selection from a normal distribution" [OED].

Random sampling was used by Karl Pearson in 1900 in the title, "On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling," Philosophical Magazine, 50, 157-175 [OED].

Random sample is found in Karl Pearson’s "On the Probable Errors of Frequency Constants," Biometrika, 2, (1903) 273: "If the whole of a population were taken we should have certain values for its statistical constants, but in actual practice we are only able to take a sample, which should if possible be a random sample." [OED].

See POPULATION.

RANDOM VARIABLE is found in 1914 in Biometrika: “nDx and nDy are now random variables independent of time.” [OED]

Variabile casuale is found in 1916 in F. P. Cantelli, “La Tendenza ad un limite nel senso del calcolo delle probabilità,” Rendiconti del Circolo Matematico di Palermo, 41, 191-201 (David, 1998). A. N. Kolmogorov used the term zufällige Gröβe in the Grundbegriffe der Wahrscheinlichkeitsrechnung (1933).

Random variable is found in 1934 in A. Winter, “On Analytic Convolutions of Bernoulli Distributions,” American Journal of Mathematics, 56, 659-663 and more visibly in H. Cramér’s Random Variables and Probability Distributions (1937) (David, 1998). Other, perhaps better, terms, including chance variable in Doob Annals of Mathematical Statistics, 6, (1935), p. 160 and stochastic variable in Wald & Wolfowitz, Annals of Mathematical Statistics, 10, (1939), p. 106 did not survive. J. L. Doob recalled the time when he was writing Stochastic Processes and W. Feller was writing his Introduction to Probability Theory and its Applications:

I had an argument with Feller. He asserted that everyone said “random variable” and I asserted that everyone said “chance variable.” We obviously had to use the same name in our books, so we decided the issue by a stochastic procedure. That is, we tossed for it and he won.

From “A Conversation with Joe Doob,” Statistical Science 1997 (p. 307) Project Euclid.

RANDOM WALK. Karl Pearson posed "The Problem of the Random Walk," in the July 27, 1905, issue of Nature (vol. LXXII, p. 294). "A man starts from a point O and walks l yards in a straight line; he then turns through any angle whatever and walks another l yards in a second straight line. He repeats this process n times. I require the probability that after these stretches he is at a distance between r and r + dr from his starting point O." Pearson’s objective was to develop a mathematical theory of random migration. In the next issue (vol. LXXII, p. 318) Lord Rayleigh translated the problem into one involving sound, "the composition of n iso-periodic vibrations of unit amplitude and of phases distributed at random," and reported that he had given the solution for large n in 1880.

This entry was contributed by John Aldrich. See RAYLEIGH DISTRIBUION.

The term RANDOMIZATION appears in 1926 in R. A. Fisher, "The Arrangement of Field Experiments", Journal of the Ministry of Agriculture of Great Britain, 33, 503-513 (David, 1995). Fisher had already described the technique in chapter VIII, section 48, Technique of Plot Experimentation, of his Statistical Methods for Research Workers (1925).

See BLOCK.

The term RANDOMIZATION TEST appears in G. E. P. Box & S. L. Andersen "Permutation Theory in the Derivation of Robust Criteria and the Study of Departures from Assumption," Journal of the Royal Statistical Society. Series B, 17, (1955), p. 3. They give the term as an alternative to PERMUTATION TEST.

(David 2001)

RANDOMIZED RESPONSE. The term and the technique were introduced by Stanley L. Warner “Randomized Response: A Survey Technique for Eliminating Evasive Answer Bias,” Journal of the American Statistical Association, 60, (1965), 63-69.

RANGE (in statistics) is found in 1848 in H. Lloyd, "On Certain Questions Connected with the Reduction of Magnetical and Meteorological Observations," Proceedings of the Royal Irish Academy, 4, 180-183 (David, 1995).

RANGE (of a function). In 1865, The Differential Calculus by John Spare has: "It is useful to become acquainted with the methods of fully examining the entire history of a function of one or more variables, in respect to the range of values which the function and its variable may sustain, and to their mutual dependence" [University of Michigan Digital Library].

Range (of a function) is found in 1914 in A. R. Forsyth, Theory of Functions of Two Complex Variables iii. 57: "A restricted portion of a field of variation is called a domain, the range of a domain being usually indicated by analytical relations" [OED].

RANK (of a determinant or matrix) was coined by F. G. Frobenius, who used the German word Rang in his paper "Uber homogene totale Differentialgleichungen," J. reine angew. Math. Vol. 86 (1879) p.1. This is according to C. C. MacDuffee, The Theory of Matrices, Springer (1933). Frobenius was defining the rank of a determinant but the term travelled.  

In English, rank (of a matrix) is found in the monograph "Quadratic forms and their classification by means of invariant factors", by T. J. Bromwich, Cambridge UP, 1906. This citation was provided by Rod Gow, who writes that it is possible that an earlier book c. 1900 by G. B. Mathews, a revision of R. F. Scott’s 1880 book on determinants, contains the word.

Rank is also found in 1907 in Introduction to Higher Algebra by Maxime Bôcher: where it is defined in the same way as in Frobenius:

Definition 3. A matrix is said to be of rank r if it contains at least one r-rowed determinant which is not zero, while all determinants of order higher than r which the matrix may contain are zero. A matrix is said to be of rank 0 if all its elements are 0. ... For brevity, we shall speak also of the rank of a determinant, meaning thereby the rank of the matrix of the determinant.

RANK CORRELATION. In his "The Proof and Measurement of Association between Two Things," American Journal of Psychology, 15, (1904), 72-101 Charles Spearman suggested that correlation between ranks be used as a measure of dependence between variables. He developed the suggestion in "’sFoot-rule’s for Measuring Correlation," British Journal of Psychology, 2, (1906), 100-101.

Karl Pearson used the term rank correlation when he criticised Spearman’s method in "Further Methods in Correlation," Drapers’s Company Res. Mem. (Biometric Ser.) IV. (1907) p. 25: "No two rank correlations are in the least reliable or comparable unless we assume that the frequency distributions are of the same general character .. provided by the hypothesis of normal distribution. ... Dr. Spearman has suggested that rank in a series should be the character correlated, but he has not taken this rank correlation as merely the stepping stone..to reach the true correlation" [OED].

[This entry was contributed by John Aldrich.]

RAO-BLACKWELL THEOREM and RAO-BLACKWELLIZATION in the theory of statistical estimation. The "Rao-Blackwell theorem" recognises independent work by C. R. Rao (1945 Bull. Calcutta Math. Soc. 37, 81-91) and David Blackwell (1947 Ann. Math. Stat., 18, 105-110). The name dates from the 1960s for previously the theorem had been referred to as "Blackwell’s theorem" or the "Blackwell-Rao theorem." The term "Rao-Blackwellization" appears in Berkson (J. Amer. Stat Assoc. 1955) ((From David (1995).)

In an ET Interview (p. 346) Rao shares some reminiscences about getting his name attached to the result, which may reflect more generally on the practice of EPONYMY. When Rao objected to Berkson’s use of Blackwellization Berkson replied that Raoization by itself "does not sound nice." The other memory was of an exchange with D. V. Lindley who had attributed the result to Blackwell. When Rao wrote to Lindley pointing out his priority, Lindley replied, "Yes, I read your paper. Although the result was in your paper, you did not realize its importance because you did not mention it in the introduction to your paper." Rao replied, saying that it was his first full-length paper and that he did not know that the introduction is written for the benefit of those who read only the introduction and do not go through the paper!

In Russia the name Rao-Blackwell-Kolmogorov theorem is used in deference to a 1950 article by Kolmogorov.

[This article was contributed by John Aldrich.]

RATIO and PROPORTION. The Latin word ratio is usually translated "computation" or "reason." St. Augustine of Hippo (354-430) used the phrase ratio numeri in De civitate Dei, Book 11, Chapter 30. The phrase is translated "science of numbers" or "theory of numbers."

According to Smith (vol. 2, page 478), ratio "is a Latin word which was commonly used in the arithmetic of the Middle Ages to mean computation.

According to Smith (vol. 2, page 478), "To represent the idea which we express by the symbols a:b the medieval Latin writers generally used the word proportio, not the word ratio; while for the idea of an equality of ratio, which we express by the symbols a:b = c:d, they used the word proportionalitas."

In De numeris datis Jordanus (fl. 1220) wrote (in translation), "The denomination of a ratio of this to that is what results from dividing this by that," according to Michael S. Mahoney in "Mathematics in the Middle Ages."

Proportion appears in 1328 in the title of the treatise De proportionibus velocitatum in motibus by Thomas Bradwardine (1290?-1349).

Proportion appears in the titles Algorismus proportionum and De proportionibus proportionum by Nicole Oresme (ca. 1323-1382). He called powers of ratios proportiones (Cajori vol. 1, page 91).

In about 1391, Chaucer wrote in English, "Abilite to lerne sciencez touchinge noumbres & proporciouns" in Treatise on the Astrolabe [OED].

In English, the word reason was used to mean "ratio" by Chaucer and later by Billingsley in his 1570 translation of Euclid’s Elements [OED].

In 1551 Robert Recorde wrote in Pathway to Knowledge: "Lycurgus .. is most praised for that he didde chaunge the state of their common wealthe frome the proportion Arithmeticall to a proportion geometricall" [OED].

Ratio was used in English in 1660 by Isaac Barrow in Euclid: Ratio (or rate) is the mutual habitude or respect of two magnitudes of the same kind each to other, according to quantity" [OED].

RATIONAL AND IRRATIONAL. The Greek word arrhetos dates back to at least the fourth century B.C., and also appears in Plato. It originally meant "unspeakable," perhaps illustrating the Greeks’s discomfort with the idea of irrational numbers, but it later became used to mean "irrational," both in Euclid’s sense and also in how we know it to be today. The term rhetos or "rational" was later created in contrast to arrhetos. [Information from Dr. Gregory Dresden, consulting "The Discovery of Incommensurability by Hippasus of Metapontum" by Kurt von Fritz and A Greek-English Lexicon by Liddell and Scott.]

According to G. A. Miller in Historical Introduction to Mathematical Literature (1916):

It shoud be noted that Euclid employed the terms rational [Greek spelling] and irrational [Greek spelling] with somewhat different meanings from those now assigned to them as defined at the beginning of this section. To explain the meaning assigned to these terms by Euclid, let a and b be rational numbers in the modern sense, and suppose that b is not a perfect square. According to Euclid’s definition the [sqrt b] is rational but a + [sqrt b] is irrational. That is, while the side of a square whose area is commensurable is incommensurable in length, Euclid says that this side is commensurable in power and considers it as rational.

Cajori (1919, page 68) writes, "It is worthy of note that Cassiodorius was the first writer to use the terms ’srational’s and ’sirrational’s in the sense now current in arithmetic and algebra."

Irrational is used in English by Robert Recorde in 1551 in The Pathwaie to Knowledge: "Numbres and quantitees surde or irrationall."

The first citation of rational in the OED2 is by John Wallis in 1685 in Alg.: "A Fraction (in Rationals) less than the proposed (Irrational) p."

RATIONAL FUNCTION. Euler used the term functio fracta in his Introductio in Analysis Infinitorum (1748).

Rational function was used by Joseph Louis Lagrange (1736-1813) in "Réflexions sur la résolution algébrique des équations," Nouveaux Mémoires de l’sAcadémie Royale, Berlin, 1770 (1772), 1771 (1773). However, the term may be considerably older [James A. Landau].

Rational function is found in English in in 1831 in the second edition of Elements of the Differential Calculus (1836) by John Radford Young: "The rational function may, however, become a maximum or a minimum for more values of x than the original root; indeed, all values of x which render the rational function negative will render every even root of it imaginary; such values, therefore, do not belong to that root; moreover, if the rational function be = 0, when a maximum, the corresponding value of the variable will be inadmissible in any even root, because the contiguous values of the function must be negative" [James A. Landau].

RATIO TEST. The term Cauchy’s ratio test appears in Edward B. Van Vleck, "On Linear Criteria for the Determination of the Radius of Convergence of a Power Series," Transactions of the American Mathematical Society 1 (Jul., 1900).

RAYLEIGH DISTRIBUION. Lord Rayleigh derived this distribution as the amplitude resulting from the addition of harmonic oscillations and presented it in "On the Resultant of a Large Number of Vibrations of the Same Pitch and of Arbitrary Phase," Philosophical Magazine, 5th series, 10, (1880), 73-78. Rayleigh later referred to the paper when Karl Pearson posed the problem of the random walk. See RANDOM WALK.

RAYLEIGH QUOTIENT, RAYLEIGH-RITZ METHOD and RAYLEIGH-RITZ RATIO are among the terms alluding—sometimes obliquely—to the work of the physicists J. W. Rayleigh and Walter Ritz: see J. W. Strutt “On the Theory of Resonance,” Phil. Trans. 161, 1870, 77-118 and Theory of Sound and W. Ritz “Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik,” J. reine angew. Math. 135 (1908) 1-61.

Whittaker and Robinson’s Calculus of Observations (second edition, 1926) has a section on the Rayleigh-Ritz Method for Minimum Problems: “To solve this problem, Rayleigh in 1870 and subsequent years devised the following method, which was afterwards elaborated in a celebrated memoir by W. Ritz.” The minimisation problem solved amounts to calculating the eigenvalues associated with a differential equation.

Rayleigh quotient appears in a paper by Collatz on ODE eigenvalue problems in 1939. Rudolf Zurmühl used it for matrices in the first edition of his book Matrizen in 1950.

The use of the name REAL ANALYSIS to refer to a subject, a course or book appears to have been a response to the growing use of the term COMPLEX ANALYSIS. The popularity of the latter dates from the 1950s while real analysis entered currency in the 1960s.

See ANALYSIS and COMPLEX ANALYSIS. Real analysis terms appear in the analysis list here.

REAL NUMBER was introduced by Descartes in French in 1637. See the entry imaginary.

The term REAL PART was used by Sir William Rowan Hamilton in an 1843 paper. He was referring to the vector and scalar portions of a quaternion [James A. Landau].

Real part also occurs in 1846 in W. R. Hamilton, Phil. Mag. XXIX. 26: "The algebraically real part may receive, according to the question in which it occurs, all values contained on the one scale of progression of numbers 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" [OED].

RECIPROCAL appears in English in 1570 in in Sir Henry Billingsley’s translation of Euclid’s Elements: "Reciprocall figures are those, when the termes of proportion are both antecedentes and consequentes in either figure."

Reciprocal occurs in English, referring to quantities whose product is 1, in the Encyclopaedia Britannica in 1797.

The term RECTANGULAR COORDINATES occurs in 1812-16 in Playfair, Nat. Phil. (1819) II. 267: "The Sun .. and .. two planets referred to the plane of the ecliptic, each by three rectangular co-ordinates..parallel to the three axes" [OED].

Rectangular coordinates also appears in a paper published by George Green in 1828 [James A. Landau].

RECTANGULAR DISTRIBUTION. See UNIFORM DISTRIBUTION.

RECTIFY (to equate a curve with a line segment). The OED shows a use in 1673 by William Brouncker in Phil. Trans. VIII. 6150: “It was easier to infer, That, if we can Rectifie the one, we may square the other.“

In a letter to Leibniz in 1693, Newton used rectificationem.

Rectification is found in English in 1707 in Glossographia Anglicana Nova by Thomas Blount: “Rectification of Curves, in Mathematicks, is the assigning or finding a streight Line equal to a curved one.” [Greg Byrnes]

RECURSION FORMULA. Recursionsformel appears in German in 1871 in Math. Annalen IV. 113 [OED].

Recursion formula appears in English in 1905 in volume I of The Theory of Functions of Real Variables by James Pierpont [James A. Landau].

RECURSIVELY ENUMERABLE SET. According to Robert I. Soare ["Computability and Recursion," Bull. symbolic logic, vol. 2 (1996), p. 300] this term debuted in Alonzo Church’s "An unsolvable problem of elementary number theory," Amer. J. Math., vol. 58 (1936), pp. 345-363. For Soare, this is "the first appearance of ’srecursively’s as an adverb meaning ’seffectively’s or ’scomputably’s." Subsequently and in that same year the term was adopted by J. B. Rosser in another important paper ["Extension of some theorems of Gödel and Church," Jour. symbolic logic, vol. 1 (1936), pp. 87-91] - and this is probably the second occurrence of the term in the literature. It is worth mentioning also that S. C. Kleene says in his book Mathematical Logic (1967) that "Such sets were first considered in" his paper "General recursive functions of natural numbers," Math. Ann., vol. 112 (1936), pp. 727-742 - in which, in fact, the term "recursive enumeration" appears, but in connection with functions; no term for the corresponding sets is introduced. The inclusion of the empty set (neglected by Kleene, Rosser and Church) was first made by Emil Post in "Recursively enumerable sets of positive integers and their decision problems," Bull. Amer. Math. Soc., vol 50 (1944), pp. 284-316. Recursively enumerable sets are now considered "the soul of recursion theory" and Post’s paper was undoubtedly responsible for this.

[This entry was contributed by Carlos César de Araújo.]

REDUCE (a fraction) is found in English in 1579 in Stratioticos by Thomas Digges: "The Numerator of the last Fragment to be reduced" [OED].

Abbreviate is found in 1796 in Mathem. Dict.: "To abbreviate fractions in arithmetic and algebra, is to lessen proportionally their terms, or the numerator and denominator" [OED].

Some writers object to the phrase "reduce a fraction" since the fraction itself is not reduced (made smaller), although the numerator and denominator are made smaller. They sometimes prefer the phrase "simplify a fraction."

REDUCTIO AD ABSURDUM. This form of argument was used by Greek mathematicians and analysed by the Greek logicians. The proof the irrationality of √2 is in Proposition 9 of Book X of Euclid’s Elements. The work of Euclid and Aristotle is considered in Chapter IX § 6. Other Technical terms of Heath’s edition of the Elements.

Reductio ad impossibile is found in English in 1552 in T. Wilson, Rule of Reason (ed. 2) f. 56: “The other croked waye (called of the Logicians, Reductio ad impossibile) is a reduccion to that, whiche is impossible” (OED).

Reductio ad absurdum is found in 1730-6 in Bailey (folio): "Exhaustions (in Mathematics) a way of proving the equality of two magnitudes by a reductio ad absurdum; shewing that if one be supposed either greater or less than the other, there will arise a contradiction" (OED).

REFLEX ANGLE. An earlier term was re-entering or re-entrant angle.

Re-entering angle appears in Phillips in 1696: "Re-entering Angle, is that which re-enters into the body of the place" [OED].

Re-entrant angle appears in 1781 in Travels Through Spain by Sir John T. Dillon: "He could find nothing which seemed to confirm the opinion relating to the salient and reentrant angles" [OED].

The 1857 Mathematical Dictionary and Cyclopedia of Mathematical Science has re-entering angle: "RE-ENTERING ANGLE of a polygon, is an interior angle greater than two right angles."

Reflex angle appears in 1876 in Syllabus of Plane Geometry, 2nd ed., by the Association for the improvement of geometrical teaching: "A reflex angle is a term sometimes used for a major-conjgate angle." [Google print search]

REFLEXIVE. (Of a binary relation) The OED cites Bertrand Russell writing in 1903 Principles of Mathematics xix. p. 159 "All kinds of equality have in common the three properties of being reflexive, symmetrical, and transitive."

REGRESSION. The statistical concept of regression has its origins in an attempt by Francis Galton (1822-1911) to find a mathematical law for one of the phenomena of heredity. His model (as it would be called today) was extended by Karl Pearson and G. Udny Yule and the biological reference eventually disappeared. The Pearson-Yule notion of regression was based on the multivariate normal distribution but R. A. Fisher re-founded regression using the model Gauss had proposed for the theory of errors and method of least squares. The Pearson-Yule and Gauss-Fisher notions are still in use. See BIOMETRY and ERROR.

The phenomenon Galton wanted to capture was reversion. Reversion occurs when an offspring resembles an ancestor more than its immediate parents. The state of knowledge on the subject in Galton’s time is summarised in chapter XIII of Charles Darwin’s Variation of Animals and Plants under Domestication (first edition 1869).

Galton’s model appears in the Appendix (p. 532) to his "Typical laws of heredity," Nature 15 (1877), 492-495, 512-514, 532-533. Galton here focussed on the inheritance of measurable characteristics; his observations are on the weight of peas. The key idea is that the offspring does not inherit all the peculiarities of the parents but is pulled back to the average of its ancestors. The idea is expressed in what would now be called a stable first-order normal autoregressive process where "time" is measured in generations. The process is stable because the reversion coefficient is the fraction of the parental deviation that is inherited.

Galton continued the search for typical laws of heredity and inspired Karl Pearson to do likewise. After the ’srediscovery’s of Mendel in 1900 much effort went into reconciling their statistical laws with his theory. Several writers contributed but the decisive reconciliation was R.A. Fisher’s The Correlation Between Relatives on the Supposition of Mendelian Inheritance Transactions of the Royal Society of Edinburgh, 52, (1918), 399-433.

Returning to regression, Galton adopted that term in 1885. See his Presidential address, Section H, Anthropology. Porter (p. 289) remarks that the significance of the change "is not entirely clear". He goes on:

Possibly [Galton] simply felt that ["regression"] expressed more accurately the fact that offspring returned only part way to the mean. More likely, the change reflected his new conviction, first expressed in the same papers in which he introduced the term "regression," that this return to the mean reflected an inherent stability of type, and not merely the reappearance of remote ancestral gemmules.

The coefficient of regression may be defined as the ratio of the mean deviation of the fraternity from the mean off-spring to the deviation of the parentage from the mean parent. ... From this special definition of regression in relation to parents and offspring, we may pass to a general conception of regression. Let A and B be two correlated organs (variables or measurable characteristics) in the same or different individuals, and let the sub-group of organs B, corresponding to a sub-group of A with a definite value a, be extracted. Let the first of these sub-groups be termed an array, and the second a type. Then we define the coefficient of regression of the array on the type to be the ratio of the mean-deviation of the array from the mean B-organ to the deviation of the type a from the mean A-organ.

The terminology and some of the symbols associated with regression appeared around this time. Pearson (1896) used the phrase "double regression" but "multiple regression coefficients" appears in the 1903 Biometrika paper "The Law of Ancestral Heredity" by Karl Pearson, G. U. Yule, Norman Blanchard, and Alice Lee.

The Pearson-Yule notion of regression which appears with correlation as an aspect of the multivariate normal specification is still current but so is a second notion developed in the 1920s. R. A. Fisher’s regression specification, where the dependent variable is normally distributed conditional on the values of the independent variables, is closer to that underlying Gauss’s first theory of least squares (see entry on Gaussian). The new regression specification appears in section 6 (p. 607) of Fisher’s 1922 "The goodness of fit of regression formulae, and the distribution of regression coefficients" (J. Royal Statist. Soc., 85, 597-612), but then to much greater effect in Statistical Methods for Research Workers. (1925). The opening paragraph of Section 15 on "Regression coefficients" (chapter V) has the declaration:

The idea of regression is usually introduced in connection with the theory of correlation, but it is in reality a more general, and, in some respects, a simpler idea, and the regression co-efficients are of interest and scientific importance in many classes of data where the correlation coefficient, if used at all, is an artificial concept of no real utility.

REGULAR (as in regular polygon) is found in 1679 in Mathematicks made easier: or, a mathematical dictionary by Joseph Moxon, with this definition: "Regular Figures are those where the Angles and Lines or Superficies are equal." The phrase "regular curve" occurs in 1665 [OED].

RELAXATION, as a term in numerical analysis for a particular method of successive approximation, derives from Southwell’s work, beginning with K. N. E. Bradfield and R. V. Southwell "Relaxation Methods Applied to Engineering Problems. I. The Deflexion of Beams under Transverse Loading," Proceedings of the Royal Society of London A, 161, (1937), 155-181. (From A. S. Householder The Theory of Matrices in Numerical Analysis (1964, p. 92).)

Gauss proposed a form of relaxation, which led G. E. Forsythe to write, "Gauss was very fond of relaxation ... [He] remarked that the process was so easy he could it do while half asleep or while thinking about other things." (with T. S. Motzkin) "An Extension of Gauss’s Transformation for Improving the Condition of Systems of Linear Equations," Mathematical Tables and Other Aids to Computation, 6, (1952), p. 17.

REMAINDER. The medieval Latin writers used numerus residuus, residuus, and residua, and various other related terms (Smith vol. 2, page 132).

In English, the word was introduced by Robert Recorde, who used remayner or remainer (Smith vol. 2, page 97).

The term REMAINDER THEOREM appears in 1886 in Algebra by G. Chrystal [OED].

RENEWAL THEORY, RENEWAL EQUATION, etc. A JSTOR search found the phrase renewal theory in A. W. Brown "A Note on the Use of a Pearson Type III Function in Renewal Theory," Annals of Mathematical Statistics, 11, (1940), 448-453. Brown refers back to A. Lotka’s 1939 paper "A Contribution to the Theory of Self-Renewing Aggregates, With Special Reference to Industrial Replacement," in the same journal. See the entry SURVIVAL FUNCTION.

Within a few years this theory of demography for machines had become part of applied probability, see J. L. Doob’s "Renewal Theory From the Point of View of the Theory of Probability," Transactions of the American Mathematical Society, 63, (1948), 422-438.

REPEATING DECIMAL. Circulating decimal is found in December 1768 in the title "On the Theory of Circulating Decimal Fractions" by John Robertson in Phil. Trans. 58:207.

Recurring decimal fraction is found in December 1768 in John Robertson, "On the Theory of Circulating Decimal Fractions," Phil. Trans. 58:207: "In operations, with such recurring decimal fractions, particularly in multiplication and division, the work will either be longer than necessary, or be very inaccurate, if the numbers are not considered as circulating ones: and to come at the true results of such operations, several authors have given precise rules; and some of them have shewn the principles upon which those rules were founded."

Repeating decimal is found in 1773 in the Encyclopaedia Britannica [OED].

Repeater is found in the 1773 edition of the Encyclopaedia Britannica: "Pure repeaters take their rise from vulgar fractions whose denominator is 3, or its multiple 9" [OED].

Infinite decimal is found in 1796 in A Mathematical and Philosophical Dictionary by Charles Hutton: "Infinite Decimals, such as do not terminate, but go on without end" [OED].

REPETEND appears in 1714 in A new and compleat treatise of the doctrine of fractions, vulgar and decimal by Samuel Cunn: "The Figure or Figures continually circulating, may be called a Repetend."

REPLACEMENT SET is dated 1959 in MWCD10.

REPLICATION as a technical term in the design of experiments appears in R.A. Fisher’s "The Arrangement of Field Experiments," Journal of the Ministry of Agriculture of Great Britain, 33, (1926) p. 506: "The method [of estimating the standard error] adopted is that of replication." (OED)

The term REPUNIT was coined by Albert H. Beiler in 1966.

RESIDUAL in a least squares context appears in 1868 in Theoretical Astronomy by James Craig Watson: "In the case of a limited number of observed values of x, the residuals given by comparing the arithmetical mean with the several observations will not ... give the true errors" [OED].

RESIDUE (in complex analysis). Cauchy introduced the concept and the term résidue in his "Sur un nouveau genre de calcul analogue au Calcul infinitesimal," Exercices de Mathématiques 1826, pp. 23-35 in Oeuvres 2e sér.. Tome VI. (F. Smithies Cauchy and the Creation of Complex Function Theory.) The OED’s earliest quotation is from A. R. Forsyth Theory of Functions x. p. 223 (1893) "The sum of the residues of a doubly-periodic function relative to a fundamental parallelogram of periods is zero."

RESIDUE (in number theory). Gauss introduced the term in his Disquisitiones arithmeticae (1801, p. 9)

Si numerus a numerorum b, c differentiam metitur, b et c secundum a congrui dicuntur, sin minus, incongrui; ipsum a modulum appelamus. Uterque numerorum b, c priori in casu alterius residuum, in posteriori vero nonresiduum vocatur. [If a number a measure the difference between two numbers b and c, b and c are said to be congruent with respect to a, if not, incongruent; a is called the modulus, and each of the numbers b and c the residue of the other in the first case, the non-residue in the latter case.]

Residue has been in English since the 14^th century and as a mathematical word since the 15^th but it was originally used to mean remainder. The OED’s first reference for the word in the modern number theory sense is the report on number theory by H. J. S. Smith Rep. Brit. Assoc. Adv. Sci. 1859 I. 231.

See MODULUS and REMAINDER.

RESIDUE CLASS appears in 1948 in Number Theory and Its History by Oystein Ore: "Since these are the numbers that correspond to the same remainder r when divided by m, we say that they form a residue class (mod m)" [OED]. A JSTOR search found an article from 1915 by Edward Kircher "Group Properties of the Residue Classes of Certain Kronecker Modular Systems and Some Related Generalizations in Number Theory," Transactions of the American Mathematical Society, 16, No. 4. (Oct), p. 413. However the German term would have been coined long before.

The term RESULTANT was introduced by Bézout, according to Geschichte der Elementar-Mathematik by Karl Fink. The term was employed by Bézout in Histoire de l’sAcademie de Paris, 1764, according to Salmon in Modern Higher Algebra.

Resultant was used by Arthur Cayley in 1856 in Phil. Trans.: "The function of the coefficients, which, equalled to zero, expresses the result of the elimination..., is said to be the Resultant of the system of quantics. The resultant is an invariant of the system of quantics" [OED].

The term eliminant was suggested by De Morgan, according to Geschichte der Elementar-Mathematik by Karl Fink.

Eliminant is found in 1881 in Burnside and Panton, Theory of Equations: "The quantity R is..called their Resultant or Eliminant" [OED].

REVERSE POLISH NOTATION is found in 1962 in “Translation to and from Polish notation” by C. L. Hamblin in The Computer Journal. The article states, “Polish notation is so-called because of its extensive use in Polish logical writings since its invention by Łukasiewicz.”

RHODONEA was coined by Guido Grandi (1671-1742) "between 1723 and 1728." He used the Greek word for "rose" (Encyclopaedia Britannica, article: "Geometry").

RHOMBUS. An obsolete term for rhombus in English was lozenge, which was used by Robert Recorde in 1551 in Pathway to Knowledge: "Defin., The thyrd kind is called losenges or diamondes whose sides bee all equall, but it hath neuer a square corner" [OED].

Rhombus was first used in English in 1567 by John Maplet in A greene forest or a naturall historie,...: "Rhombus, a figure with ye Mathematicians foure square: hauing the sides equall, the corners crooked" [OED].

The term RHUMB LINE is due to Portuguese navigator and mathematician Nunes (Nonius) (Smith vol. I).

RICCATI EQUATION is the name given to a differential equation studied by Vincenzo Riccati in his “Animadversationes in aequationes differentiales secundi gradus,” Acta eruditorum Supp. viii. (1724), pp. 66-73. See Klein (ch. 21 “Ordinary Differential Equations in the Eighteenth Century”) and the entry in the Enyclopedia of Mathematics.

According to Mathematical Thought From Ancient to Modern Times by Morris Kline, the term Riccati equation was introduced by D’sAlembert. The reference is Hist. de l’sAcad. de Berlin, 19, 1763, 242 ff. pub. 1770.

An article, “Mr.Murphy’s Solution of Riccati’s Equation,” is found in 1837 in The Ladies’ Diary. The article is reprinted from the Cambridge Philosophical Society’s Transactions. [Google print search, James A. Landau]

The matrix Riccati equation came to prominence in the 1960s in connection with optimal control problems but matrix equations “of the Riccati type” had been studied in the mathematics literature since the 1930s. See W. M. Whyburn “Matrix Differential Equations, American Journal of Mathematics, 56, (1934) 587-592.

RICHARD’S PARADOX was presented as a contradiction in set theory by Jules Richard (1862-1956) in a 1905 letter, "Les principes des mathématiques et le problème des ensembles" Revue générale des sciences pures et appliquées, 16, (1905), 541 (translated in Heijenoort (1967)). The contradiction was associated with the set of numbers that can be defined in a finite number of words. Bertrand Russell referred to the contradiction as "le paradoxe du Richard" in his Les Paradoxes de la Logique Revue de métaphysique et de morale (1906) p. 638. Peano’s reaction to the contradiction was that it did not belong to mathematics but to linguistics. It is now classified as a semantic paradox.

See PARADOX.

RIEMANN HYPOTHESIS. In 1905 Euclid’s Parallel Postulate: Its Nature, Validity, and Place in Geometrical Systems by John William Withers has: "If we accept Riemann’s hypothesis we cannot be sure that there will be any such line at all, for we do not know that space has any infinitely distant parts." [Google print search]

Riemann hypothesis appears in English in its modern sense in 1924 in the Proceedings of the Cambridge Philosophical Society XXII: "We assume Riemann’s hypothesis. ... We assume the truth of the Riemann hypothesis" [OED]. The hypothesis is stated by Bernhard Riemann (1826-1866) in "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" (1859) Werke (p. 139) English translation (p. 4).

RIEMANNIAN GEOMETRY is dated 1904 in MWCD10.

RIEMANN INTEGRAL appears in 1907 in The theory of functions of a real variable and the theory of Fourier’s series by Ernest William Hobson: "The Lebesgue integral of f(x) lies between the upper and lower integrals of f(x), and is identical with the Riemann integral in case the latter exists." [University of Michigan Historical Math Collection]

The term RIEMANN SPACE was used by Alfred Clebsch in a lecture delivered on Aug. 28, 1893: "There is no difficulty in conceiving a four-dimensional Riemann space corresponding to an equation f(z, y, z) = 0." [Google print search]

RIEMANN SUM is found in 1935 in Interpolation and Approximation by Rational Functions in the Complex Domain by Joseph Leonard Walsh: "The sum in the left-hand member of (23) is precisely a Riemann sum for the integral in the right-hand member if we have x_nk = k/n. [Google print search]

RIEMANN ZETA FUNCTION. The use of the small letter zeta for this function was introduced by Bernhard Riemann (1826-1866) as early as 1857 (Cajori vol. 2, page 278).

Riemann ζ function appears in 1899 in Messenger of Mathematics XXIX. 114 [OED].

Riemannian prime number function appears in the title of H. von Koch, "Ueber die Riemann’ssche Primzahlfunction," Math. Annalen 55 (1902) 441-464 [James A. Landau].

RIESZ-FISCHER THEOREM. The theorem was proved independently by F. Riesz and E. Fischer in 1907. Riesz published “Sur les systèmes orthogonaux de fonctions,” Comptes Rendus de l’sAcadémie des Sciences, 144 (1907), 615--619 and then Fischer followed with “Sur la convergence en moyenne,” Comptes Rendus de l’sAcadémie des Sciences, 144 (1907), 1022-1024. The theorem is always known by their joint names, as in W. H. Young Proceedings of the Royal Society, 88, (1913) p. 170 “... in the light of the Riesz-Fischer theorem...” See Enyclopedia of Mathematics.

RIGHT, OBTUSE and ACUTE ANGLES are the subjects of Definitions 10, 11 and 12 of Book 1 of Euclid’s Elements. The modern English word right is derived from Old English while obtuse and acute are derived from the Latin words for blunt and sharp respectively. See the entry ANGLE.

The OED’s earliest citation for right angle in English is from Chaucer’s Treatise on the Astrolabe (c. 1391): “This forseid rihte orisonte..diuideth the equinoxial in-to riht Angles.” The earliest citation for the other words is from 1570 and Sir Henry Billingsley’s translation of Euclid’s Elements: “an obtuse angle is that which is greater then a right angle" and "An acute angle is that, which is lesse then a right angle."

RIGHT TRIANGLE. Right cornered triangle is found in 1551 in Pathway to Knowledge by Robert Recorde: "Therfore turne that into a right cornered triangle, accordyng to the worke in the laste conclusion" [OED].

Right angled triangle is found in 1594 in Exercises (1636) by Blundevil: "If they have right sides, such Triangles are eyther right angled Triangles, or oblique angled Triangles" [OED].

Right triangle is found in 1675 in R. Barclay, Apol. Quakers: "A Mathematician can infallibly know, by the Rules of Art, that the three Angles of a right Triangle, are equal to two right Angles" [OED].

The term rectangular triangle appears in 1678 in Cudworth, Intell. Syst.: "The Power of the Hypotenuse in a Rectangular Triangle is Equal to the Powers of both the Sides" [OED].

RING. Richard Dedekind (1831-1916) introduced the concept of a ring. See MacTutor The Development of Ring Theory.

The term number ring (Zahlring) was coined by David Hilbert (1862-1943) in the context of algebraic number theory [See Jahresbericht der Deutschen Mathematiker-Vereinigung Capitel IX. Die Zahlringe des Körpers.

See also “Die Theorie der algebraische Zahlkoerper,” Jahresbericht der Deutschen Mathematiker Vereiningung, Vol. 4, 1894/5] Gesammelte Abhandlungen Bd. 1.

The first axiomatic definition of a ring was given in 1914 by Adolf A. Fraenkel (1891-1965) in an essay “Über die Teiler der Null und die Zerlegung von Ringen” in Journal für die reine und angewandte Mathematik (A. L. Crelle) p. 139 vol. 145, 1914.

Ring is found in English in 1915 in Transactions of the American Mathematical Society: “The ring of the rational g-adic numbers.” (OED)

[Julio González Cabillón]

RISK and RISK FUNCTION (referring to the expected value of the loss in statistical decision theory) first appear in Wald’s "Contributions to the Theory of Statistical Estimation and Testing  Hypotheses," Annals of Mathematical Statistics,, 10, (1939), 299-326 [John Aldrich, based on David (2001)].

See DECISION THEORY.

The term ROBUSTNESS was used in a 1953 article by George E. P. Box: "This remarkable property of ’srobustness’s to non-normality which these tests for comparing means possess, and without which they would be much less appropriate to the needs of the experimenter, is not necessarily shared by other statistical tests." Box’s "Normality and Tests on Variances" (Biometrika, 40, 318-335) belonged to a literature with its origins in the 1920s when W. A. Shewhart and E. S. Pearson first investigated the behaviour of Student’s t-test and the analysis of variance under non-standard conditions. (OED2 and David 2001)

ROLLE’S THEOREM was stated by Michel Rolle in his Démonstration d’sune méthode pour résoudre les égalitéz de tous les dégrez (1691). Kline (p. 381) adds that Rolle stated the theorem but did not prove it.

According to Cajori (1919, page 224) the term Rolle’s theorem was first used in 1834 by Moritz Wilhelm Drobisch (1802-1896) and in 1846 by Giusto Bellavitis (1803-1880).  Bellavitis used teorema del Rolle in 1846 in the Memorie dell’s I. R. Istituto Veneto di Scienze, Lettere ed Arte, Vol. III (reprint), p. 46, and again in 1860 in Vol. 9, section 14, page 187.

Rolle’s theorem is found in English in 1858 in A treatise on the theory of algebraical equations by John Hymers [Univesity of Michigan Historical Math Collection].

ROMAN NUMERAL is found in 1735 in Phil. Trans. xxxix, 139: "The Roman Numeral Ten, which was made in this Form, like an X" [OED].

Also in 1735, The Whole Duty of Man According to the Law of Nature by Samuel Puffendorf has: "The Roman Numerals I and II, signifie the First and Second Book." [Google print search]

ROOT-MEAN-SQUARE is found in Sept. 1895 in Electrician: "A short time ago Dr. Fleming published a new and ingenious method of plotting wave forms with polar co-ordinates, and of directly obtaining therefrom the root mean-square value" [OED].

ROOT of an equation. See RADIX.

ROOT in graph theory. The OED gives A. Cayley in "On the Theory of Analytical Forms called Trees," Phiosophical Magazine XIII. (1857) Papers, II, p. 242  "The inspection of these figures will show at once what is meant by the term in question, and by the terms root, branches, ... and knots (which may be either the root itself, or proper knots, or the extremities of the free branches)."

ROOT TEST appears in 1937 in Differential and Integral Calculus, 2nd. ed. by R. Courant [James A. Landau].

ROTUNDUM is a Latin word introduced by Peter Ramus (1515-1572) to refer to the circle or the sphere (DSB).

The term ROULETTE was coined by Pascal (Cajori 1919, page 162). See also cycloid and trochoid.

ROUND (verb; to approximate a number) is found in English in 1840 in Observations on the attempted application of pantheistic principles to the theory and historic criticism of the gospel by William Hodge Mill: “300 years: which, after every possible allowance for rounding the number, will give..a length mostly absolutely incompatible with the supposition of only two generations.” [OED]

Round up and round down are found in 1956 in G. A. Montgomerie, Digital Calculating Machines vii. 129: "In a long calculation, all these increases may accumulate, and it is better to round some of them up and some of them down" [OED].

RULE OF FALSE POSITION. The Arabs called the rule the hisab al-Khataayn and so the medieval writers used such names as elchataym.

Fibonacci in the Liber Abaci has a heading De regulis elchatayn.

In his Suma (1494) Pacioli used el cataym.

Cardano used the term regula aurea, according to Cajori (1906).

Peletier (1549, 1607 ed., p. 269) used "Reigle de Faux, mesmes d’sune Position."

In 1551 Robert Recorde in Pathway to Knowledge wrote: "Also the rule of false position, with dyuers examples not onely vulgar, but some appertaynyng to the rule of Algeber" [OED].

Trenchant (1566; 1578 ed., p 223) used "La Reigle de Faux."

Baker (1568; 1580 ed., fol. 181) used "Rule of falshoode, or false positions" (Smith vol. 2, page 438).

Suevus (1593, p. 377) used "Auch Regula Positionum genant."

RULE OF SUCCESSION. This states that if an event has occurred n times in succession, then the probability that it will occur again is (n + 1)/(n + 2). This value is the mean of the posterior density of the probability of a success given n successes in n Bernoulli trials assuming a uniform prior for the probability of a success.

John Venn introduced the expression "rule of succession" in his Logic of Chance (first edition 1865) and he spent a chapter criticising the rule. It is often called Laplace‘s rule of succession because it appears in the Introduction Laplace added to the 2^nd (1814) edition of his Théorie Analytique des Probabilités. Laplace also published the introduction separately as the Essai Philosophique sur les Probabilités.

Laplace’s illustration (p. xvii in the edition on Gallica) became famous, "if we place the dawn of history at five thousand years before the present date, we have 1,826,213 days on which the sun has constantly risen in each 24 hour period. We may therefore lay odds of 1,826,214 to one that it will rise again tomorrow." However, Laplace continued, "But this number would be incomparably greater for one, who perceiving in the coherence {or totality} of phenomena the principle regulator of days and seasons, sees that nothing at the present moment can arrest the course of it." (from A. I. Dale’s translation of the Essai, Pierre-Simon Laplace: Philosophical Essay on Probabilities (p. 11, 1995).

Earlier Richard Price in his Appendix to Bayes’s An Essay towards solving a Problem in the Doctrine of Chances (describing applications) had discussed another version of the problem of the naive person wondering whether the sun would keep on rising (p. 19 in the web version).

See the entry on Bayes.

(This entry was written by John Aldrich, based on Dale op. cit. and two articles by S. L. Zabell (1988): "Buffon, Price and Laplace: Scientific Attribution in the 18^th century," Archive for History of Exact Sciences, 39, 173-181 and (1989) "The Rule of Succession," Erkenntnis, 31, 283-321.)

The term RULE OF THREE was used by Brahmagupta (c. 628) and by Bhaskara (c. 1150) (Smith vol. 2, page 483).

From Smith (vol. 2, pp. 484-486):

Robert Recorde (c. 1542) calls the Rule of Three "the rule of Proportions, whiche for his excellency is called the Golden rule," although his later editors called it by the more common name. Its relation to algebra was first strongly emphasized by Stifel (1553-1554). When the rule appeared in the West, it bore the common Oriental name, although the Hindu names for the special terms were discarded. So highly prized was it among merchants, however, that it was often called the Golden Rule, a name apparently in special favor with the better mathematical writers. Hodder, the popular English arithmetician of the 17th century, justifies this by saying: "The Rule of Three is commonly called, The Golden rule; and indeed it might be so termed; for as Gold transcends all other mettals, so doth this Rule all others in Arithmetick." The term continued in use in England until the end of the 18th century at least, perhaps being abandoned because of its use in the Church.

Rule of three appears in English in 1562 in Well Sprynge Science by H. Baker: “The rule of three is the chiefest, the moste profitable, and the moste excellente rule of all the rules of Arithmetike.” [OED]

Numerous 18th- and 19th-century wills and other documents which can be found on the Internet require that certain persons should learn arithmetic "to the rule of three."

Abraham Lincoln (1809-1865) used rule of three in an autobiography he wrote on December 20, 1859:

There were some schools, so called; but no qualification was ever required of a teacher beyond "readin, writin, and cipherin" to the Rule of Three. If a straggler supposed to understand latin happened to sojourn in the neighborhood, he was looked upon as a wizzard. There was absolutely nothing to excite ambition for education. Of course when I came of age I did not know much. Still somehow, I could read, write, and cipher to the Rule of Three; but that was all. I have not been to school since. The little advance I now have upon this store of education, I have picked up from time to time under the pressure of necessity.

I have no faith in anything short of actual measurement and the Rule of Three.

The term RUNGE-KUTTA METHOD (named for Carle Runge and Wilhelm Kutta) apparently was used by Runge himself in 1924, according to Chabert (p. 441), who writes:

Notons que dans l’souvrage de Runge et König de 1924, la méthode à laquelle Kutta a abouti est appellé méthode de Runge-Kutta ([19], p. 286.

The bibliography quote is: [19] C. Runge et H. König, Vorlesungen über numerisches Rechnen, Springer, Berlin, 1924. [This information was provided by Manoel de Campos Almeida.]

RUSSELL’S PARADOX. Bertrand Russell (1872-1970) found the contradiction (as he called it) in 1901. Chapter X of his Principles of Mathematics (1903) is called "The Contradiction." Ernst Zermelo (1871-1953) discovered the paradox in 1899 but he did not publish it. See Grattan-Guinness (2000, p. 216)

"Der Russellschen Antimonie" appears in Ernst Zermelo " Untersuchungen über die Grundlagen der Mengenlehre. I" Math. Annalen (1908, p. 261) The OED’s earliest English quotation is from a passage in the 1922 translation of Ludwig Wittgenstein’s Tractatus Logico-Philosophicus p. 57: "Herewith Russell’s paradox vanishes."