*Last revision: Dec. 20, 2013*

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

An isolated matter of interest is the origin of the term 'radian', 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 'rad', 'radial' and 'radian'. In 1874, T. Muir adopted 'radian' after a consultation with James Thomson.In a footnote, Cajori gives a reference to

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 'radian,' on the ground that it
could be viewed as a contraction for 'radial angle'..."

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
'radian.' ...It thus appears that 'radian' 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.This work had earlier been translated by Thomas Branker (1636-1676), from the original by J. H. Rahn, first published in 1659 in German.

**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"
(OED2).

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" (OED2).

*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" (OED2).

*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" (OED2).

*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.[Oeuvres v.1, p. 186-7, v.3, p. 161-2]

*Unknown* is found in English in 1676 in Glanvill, *Ess.*:
"The degree of Composition in the unknown Quantity of the Æquation"
(OED2).

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 Navigation*by 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 'random distribution' 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** is found in Harald Cramér, "Random
variables and probability distributions," *Cambridge Tracts in
Math. and Math. Phys.* 36 (1937).

**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 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" (OED2).

*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 (OED2).

*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."
(OED2).

See POPULATION.

**RANDOM VARIABLE.** *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"
(OED2).

**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 "'Foot-rule' 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'
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" (OED2).

[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* (OED2).

In English, the word *reason* was used to mean "ratio" by
Chaucer and later by Billingsley in his 1570 translation of
Euclid's *Elements* (OED2).

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" (OED2).

*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"
(OED2).

**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' 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, letaandbbe rational numbers in the modern sense, and suppose thatbis not a perfect square. According to Euclid's definition the [sqrt b] is rational buta+ [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 commensurablein powerand considers it as rational.

Cajori (1919, page 68) writes, "It is worthy of note that Cassiodorius was the first writer to use the terms 'rational' and 'irrational' 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'Académie 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.

Extensions of the Rayleigh method appear in the
numerical linear algebra literature around 1950. A *JSTOR* search for the term **Rayleigh
quotient** found S. H. Crandall “Iterative Procedures Related to
Relaxation Methods for Eigenvalue Problems,” *Proceedings
of the Royal Society, A*, **207**,
(1951), 416-423. Crandall uses the phrase “Rayleigh’s quotient” and describes a
method which he describes as a slight extension of that used by Rayleigh.

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"
(OED2).

**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" (OED2).

*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 (OED2).

*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
'recursively' as an adverb meaning 'effectively' or 'computably'."
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" (OED2).

*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" (OED2).

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" (OED2).

*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"
(OED2).

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 'rediscovery' 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.For Galton, regression/reversion, unlike correlation (q.v.) another of his inventions, existed only in this biological context. The same might have been true for Karl Pearson. However Pearson's first major paper on heredity ("Regression, Heredity, and Panmixia,"

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 biological terminology of "organs" etc. had gone when G. U. Yule presented his boss's theory in "On the Theory of Correlation" (

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 dependent 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.This entry was contributed by John Aldrich, drawing on David (1995) and on the description of Galton's work in Porter and of the Galton-Pearson-Yule development in Stigler (1986). See also AUTOREGRESSION, CORRELATION, DEPENDENT VARIABLE, DUMMY VARIABLE, GAUSS-MARKOV THEOREM, METHOD OF LEAST SQUARES, NORMAL EQUATION, RESIDUAL, WEIGHT and the discussion of regression notation on the Symbols in Probability and Statistics page.

**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 (OED2).

**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' 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 (OED2).

**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* (OED2).

*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" (OED2).

*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" (OED2).

**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" (OED2).

**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 numerusanumerorumb, cdifferentiam metitur,betcsecunduma congruidicuntur, sin minus,incongrui; ipsuma modulumappelamus. Uterque numerorumb, cpriori in casu alteriusresiduum,in posteriori verononresiduumvocatur. [If a numberameasure the difference between two numbersbandc, bandcare said to be congruent with respect toa,if not, incongruent;ais called the modulus, and each of the numbersbandcthe 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*)" (OED2). 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'Academie 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" (OED2).

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" (OED2).

**REVERSE POLISH NOTATION.** According to a number of web pages,
“Prefix notation also came to be known as Polish Notation in
honor of Lukasiewicz.
HP adjusted the postfix notation for a calculator keyboard, added a
stack to hold the operands and functions to reorder the stack. HP
dubbed the result Reverse Polish
Notation (RPN) also in honor of Lukasiewicz.”

A *JSTOR* search finds the term in use in 1973, but it may be older.

**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" (OED2).

*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" (OED2).

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'Alembert.
The reference is Hist. de l'Acad. 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" (OED2). 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}* =

**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'sche 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'Académie des Sciences*, 144 (1907), 615--619
and then Fischer followed with “Sur la convergence en moyenne,”
*Comptes Rendus de l'Académie 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" (OED2).

*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" (OED2).

*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" (OED2).

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" (OED2).

**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 1930 in E. T. Bell, “Rings whose
elements are ideals,” *Bulletin A. M. S.*

[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 'robustness' 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'une 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' 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" (OED2).

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"
(OED2).

**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 *Webster's
New International Dictionary,* 2nd ed. (1934).

*Round* is found in 1935 in Shuster and Bedford, *Field Work
in Math.*: "Round the following numbers to three significant
figures" (OED2).

*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" (OED2).

**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'une 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" (OED2).

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.

*Golden Rule* is found in 1702 in Joseph Raphson's *Mathematical Dictionary*:
"*Golden Rule,* otherwise called the Rule of Three, teaches to
find a fourth Proportional to three Numbers given; it may be either
*Direct* or *Inverse.*" This dictionary also has:
"This Multiplication commonly happens in the Rule of Three, and also in Practical Geometry
for measuring Plains and Solids."

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.Charles Darwin (1809-1882) wrote:

I have no faith in anything short of actual measurement and the Rule of Three.[The Darwin quotation was provided by John Aldrich, who points out the interesting fact that Lincoln and Darwin were born on the same day, February 12, 1809.]

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'ouvrage 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."