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).
RADIOGRAM appears in a syllabus prepared by Karl Pearson in 1892, according to Stigler [James A. Landau].
RADIUS. 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."
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 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."
The term radius of curvature may have been used earlier by Christiaan Huygens and Isaac Newton, who wrote on the subject.
The term RADICAL SIGN is found in English in An Introduction to Algebra edited in 1668 by John Pell (1611-1685). This work had earlier been translated by Thomas Branker (1636-1676), from the original by J. H. Rahn, first published in 1659 in German. The word radical had been used in English before 1668 by Recorde and others to refer to an irrational number.
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).
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 1924 in Bell System Technical Journal III 88 (OED2).
RANDOM SAMPLING is found in 1900 in Philosophical Magazine (OED2).
RANDOM SELECTION occurs in 1898 in Philosophical Transactions of the Royal Society (OED2).
RANDOM VARIABLE is found in Harald Cramér, "Random variables and probability distributions," Cambridge Tracts in Math. and Math. Phys. 36 (1937).
The term RANDOM WALK was coined by Karl Pearson in the brief letter, "The Problem of the Random Walk," published in the July 17, 1905, issue of Nature [James A. Landau; DSB].
RANGE (of a function) is used in 1914 by J. G. Kemeny et al. in Finite Mathematical Structures (OED2).
RANK (of a matrix) is found in "General Theory of Modular Invariants," Leonard Eugene Dickson, Transactions of the American Mathematical Society, Vol. 10, No. 2. (Apr., 1909).
RATIO. 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. 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."
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).
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."
The term 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].
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].
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 appears in a paper published by George Green in 1828 [James A. Landau].
RECURRING DECIMAL is found in the 1801 supplement to the 1797 Encyclopaedia Britannnica (OED2).
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.
An earlier term for a reflex angle, especially as part of a polygon, is re-entrant or re-entering angle.
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."
In an 1877 lecture Galton used the term reversion coefficient.
The OED2 shows a use by Karl Pearson of coefficient of regression in 1897 in Phil. Trans. R. Soc. (OED2).
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 is found in 1773 in the Encyclopaedia Britannica (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.
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).
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).
RHODONEA was coined by Guido Grandi (1671-1742) "between 1723 and 1728." He used the Greek word for "rose" (Encyclopaedia Britannica, article: "Geometry").
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 (OED2).
RIEMANNIAN GEOMETRY is dated 1904 in MWCD10.
RIEMANN INTEGRAL appears in 1914 in the Proceedings of the London Mathematical Society:
Corresponding to Riemann's extension of the notion of an integrable function, we now have a certain class of functions which may be said to possess a "Riemann" integral with respect to the monotone increasing function g(x), that is to say a function such that the summation...has a unique and finite limit, however the points x are chosen in their corresponding intervals, and however those intervals are constructed, provided only the length of the greatest of them approaches zero as n ->(This citation is from the OED2).
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).
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 is found in Chauvenet, A Treatise on Elementary Geometry (1870): "A right triangle is one which has a right angle; as MNP, which is right-angled at N." [Tom Foregger].
Right angled triangle appears in Elements of Plane Geometry, Part I, with an appendix on Mensuration by Thomas Hunter (1871) [Tom Foregger].
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]. Richard Dedekind (1831-1916) was first to introduce the concept of a ring. 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. (This entry contributed by Julio González Cabillón.)
ROLLE'S THEOREM. According to Cajori (1919, page 224) the term was first used in 1834 by Moritz Wilhelm Drobisch (1802-1896).
According to the DSB (article: "Rolle"), the term was first used in 1846 by Giusto Bellavitis (1803-1880).
ROMAN NUMERAL is found in 1735 in Phil. Trans. xxxix, 139 (OED2).
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 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).
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.
Peletier (1549, 1607 ed., p. 269) used "Reigle de Faux, mesmes d'une Position."
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 the term in a 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.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 (, p. 286.The bibliography quote is:  C. Runge et H. König, Vorlesungen über numerisches Rechnen, Springer, Berlin, 1924. [This information was provided by Manoel de Campos Almeida.]