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

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 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].

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 appears in a syllabus prepared by Karl Pearson in 1892, according to Stigler [James A. Landau].

RADIUS. Archimedes called the radius "ek tou kentrou" (the [line] from the center) [Samuel S. Kutler].

The term "radius" was not used by Euclid, the term "distance" being thought sufficient (Smith vol. 2, page 278).

According to Smith (vol. 2, page 278), Boethius (c. 510) seems to have been the first to use the equivalent of our "semidiameter."

Semidiameter appears in English 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" (OED2).

Radius 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).

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]. The term may be considerably older.

In Differential and Integral Calculus (1902) by Virgil Snyder and J. I. Hutchinson, the authors use interval of convergence.

In Differential and Integral Calculus (1908), Daniel A. Murray uses interval of convergence.

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.

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."

The term radix, meaning a number which is the basis of a scale of numeration, is due to Robert Flowers in 1771, according to A. J. Ellis in Nature (1881) XXIII. 379/2 (OED2).

In the 1939 movie The Wizard of Oz, when the Scarecrow receives his brain, he 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."

RANDOM DISTRIBUTION is found in L. S. Ornstein, "Mean values of the electric force in a random distribution of charges," Proc. Akad. Wet. Amsterdam 38 (1935).

RANDOM NUMBER. The phrase "this table of random numbers" is found in 1927 in Tracts for Computers (OED2).

See also L. H. C. Tippett, "Random Sampling Numbers 1927," Tracts for Computers, No. 15 (1927) [James A. Landau].

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 April 1870 in "Notices of Recent Publications," The Princeton review: "We confess that we have never suspected Satan as capable of poetizing in the manner attributed to him in Book IX, of which the following is a random sample."

Random choice appears in the Century Dictionary (1889-1897).

Random selection is found in 1868 in The American Wheat Culturist by Sereno Edwards Todd: "Next year the best head from the first-mentioned ear was planted as before. From this the best grain produced 21 heads, containing from 91 to 55 grains per head, or in all 1,190. The best random head of the other ear was also planted; but it was thrown out as being evidently inferior to the others. From this, Hallett deduces the first proof of the correctness of his idea that careful breeding and cultivation was correct, and not the random selection of good specimens" [University of Michigan Digital Library].

Random selection occurs in 1897 in Proc. R. Soc. LXII. 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 1903 in Biometrika II. 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).

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).

Random variable is found in 1934 in Aurel Freidrich Wintner, "On Analytic Convolutions of Bernoulli Distributions," American Journal of Mathematics, 56, 659-663 (David, 1998).

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 [John Aldrich].

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).

According to Tankard (p. 112), R. A. Fisher "may ... have coined the term randomization; at any rate, he certainly gave it the important position in statistics that it has today."

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 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) pp.1-19; also in Collected Works of FGF, Vol I, p. 435. This is according to C. C. MacDuffee, The Theory of Matrices, Springer (1933).

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:

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 appears in 1907 in Drapers' Company Res. Mem. (Biometric Ser.) IV. 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).

RAO-BLACKWELL THEOREM and RAO-BLACKWELLIZATION in the theory of statistical estimation. The "Rao-Blackwell theorem" recognises independent work by Rao (1945 Bull. Calcutta Math. Soc. 37, 81-91) and 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).)

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. 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.

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).

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 occurs in G. A. Carlton, "Estimating the parameters of a rectangular distribution," Ann. Math. Statis. 17 (1946) [James A. Landau].

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. 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" (OED2).

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" (OED2).

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 is defined in 1889 in the Century Dictionary [Mark Dunn]. It also appears in the 1913 edition of Plane and Solid Geometry by George A. Wentworth, and may occur in the earliest edition of 1888, which has not been consulted.

REGRESSION. According to the DSB, Francis Galton (1822-1911) discovered the statistical phenomenon of regression and used this term, although he originally termed it "reversion."

Porter (page 289), referring to Galton, writes:

He did, however, change his terminology from "reversion" to "regression," a shift whose significance is not entirely clear. Possibly he simply felt that the latter term 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.

Galton used the term reversion coefficient in "Typical laws of heredity," Nature 15 (1877), 492-495, 512-514 and 532-533 = Proceedings of the Royal Institution of Great Britain 8 (1877) 282-301.

Galton used regression in a genetics context in "Section H. Anthropology. Opening Address by Francis Galton," Nature, 32, 507-510 (David, 1995).

Galton also used law of regression in 1885, perhaps in the same address.

Karl Pearson used regression and coefficient of regression in 1897 in Phil. Trans. R. Soc.:

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 phrase "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. From around 1895 Pearson and Yule had worked on multiple regression and the phrase "double regression" appears in Pearson's paper "Mathematical Contributions to the Theory of Evolution. III. Regression, Heredity, and Panmixia" (Phil. Trans. R. Soc. 1896). [This paragraph was contributed by John Aldrich.]

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).

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).

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).

REPETEND appears in 1714 in Treat. Fractions by Cunn: "The Figure or Figures continually circulating, may be called a Repetend."

REPLACEMENT SET is dated 1959 in MWCD10.

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 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).

The term RESULTANT was employed by Bezout, 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 introduced I think by Professor De Morgan," according to Salmon in Modern Higher Algebra.

Eliminant is found in 1881 in Burnside and Panton, Theory of Equations: "The quantity R is..called their Resultant or Eliminant" (OED2).

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).

The term RICCATI EQUATION was introduced by D'Alembert (Kline, page 484).

RIEMANN HYPOTHESIS appears in English 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).

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 [University of Michigan Historical Math Collection].

The term RIEMANN SPACE was used by Poul Heegaard in 1898 in Forstudier til en topologisk teori for de algebraiske fladers sammenhoeng.

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).

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].

An early use in English of the term Riemann zeta function occurs in "Some Asymptotic Expressions in the Theory of Numbers," T. H. Gronwall, Transactions of the American Mathematical Society 14 (Jan., 1913).

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.

The term ring (Zahlring) was coined by David Hilbert (1862-1943) in the context of algebraic number theory [See "Die Theorie der algebraische Zahlkoerper," Jahresbericht der Deutschen Mathematiker Vereiningung, Vol. 4, 1897].

The first axiomatic definition of a ring was given in 1914 by A. A. Fraenkel (1891-1965) in an essay in Journal fuer die reine und angewandte Mathematik (A. L. Crelle), 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)].

ROLLE'S THEOREM. According to Cajori (1919, page 224) the term 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 (OED2).

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 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."

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.

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 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.

Runge-Kutta method appears in 1930 in J. B. Scarborough, Numerical Math. Analysis: "In the special case where dy/dx is a function of x alone the Runge-Kutta method reduces to Simpson's rule" (OED2).