Earliest Known Uses of Some of the Words of Mathematics

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 Nature, Vol. 83, pp. 156, 217, 459, 460 [Julio González Cabillón].

In a letter appearing in the April 7, 1910, Nature, Thomas Muir wrote: "I wrote to him [i.e., to Alexander J. Ellis, in 1874], and he agreed at once for the form '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."

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 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.

[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."

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 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) 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. Kendall & Stuart vol ii page 494 say that the rank correlation coefficient was introduced by "the eminent psychologist" Spearman in 1906. Pearson's biography of Galton also uses the term "correlation of ranks" [James A. Landau].

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.

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."

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.

In 1859 Charles Darwin used reversion in a biological context in The Origin of Species (1860): "We could not have told, whether these characters in our domestic breeds were reversions or only analogous variations" (OED2).

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.

[OED2]

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.

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.

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 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.]

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).

ST. ANDREW'S CROSS is the term used by Florian Cajori for the multiplication symbol X. It appears in 1916 in his "William Oughtred, A Great Seventeenth-Century Teacher of Mathematics.

St. Andrew's cross is found in 1615, although not in a mathematical context, in Crooke, Body of Man: "[They] doe mutually intersect themselues in the manner of a Saint Andrewes crosse, or this letter X" (OED2).

The term ST. PETERSBURG PARADOX was coined by d'Alembert, who received a solution by Daniel Bernoulli in 1731 and published it in Commentarii Akad. Sci. Petropolis 5, 175-192 (1738). The originator of the St. Petersburg paradox was Niklaus Bernoulli. (Jacques Dutka, "On the St. Petersburg paradox," Arch. Hist. Exact Sci. 39, No.1, 1988)

SADDLE POINT is found in 1922 in A Treatise on the Theory of Bessel Functions by G. N. Watson (OED2).

SAGITTA was used in Latin by Fibonacci (1220) to mean the versed sine (Smith, vol. 2). See versed sine.

In 1726 Alberti's Archit. has: "The .. Line .. from the middle Point of the Chord up to the Arch, leaving equal Angles on each Side, is call'd the Sagitta" (OED2).

Webster's New International Dictionary (1909) has the following definition for sagitta: "the distance from a point in a curve to the chord; also, the versed sine of an arc; -- so called (by Kepler) from its resemblance to an arrow resting on the bow and string; also, Obs., an abscissa.

The 1961 third edition of the same dictionary has the following definition: "the distance from the midpoint of an arc to the midpoint of its chord."

SALIENT ANGLE. The OED2 has a 1687 citation for Angle Saliant.

In 1781 Sir John T. Dillon wrote in Travels Through Spain: "He could find nothing which seemed to confirm the opinion relating to the salient and reentrant angles" (OED2).

Mathematical Dictionary and Cyclopedia of Mathematical Science (1857) has: "SALIENT ANGLE of a polygon, is an interior angle, less than two right angles."

See also rank correlation.

SIGNIFICANT DIGIT. Smith (vol. 2, page 16) indicates Licht used the term in 1500, and shows a use of "neun bedeutlich figuren" by Grammateus in 1518.

In 1544, Michael Stifel wrote, "Et nouem quidem priores, significatiuae uocantur."

Signifying figures is found in 1542 in Robert Recorde, Gr. Artes (1575): "Of those ten one doth signifie nothing... The other nyne are called Signifying figures" (OED2).

Significant figures is found in 1660 in Milton, Free Commw.: "Only like a great Cypher set to no purpose before a long row of other significant Figures" (OED2).

Significant figures is found in the first edition of the Encyclopaedia Britannica (1768-1771) in the article "Arithmetick": "Of these, the first nine, in contradistinction to the cipher, are called significant figures."

Mathematical Dictionary and Cyclopedia of Mathematical Science (1857) has this definition:

SIGNIFICANT. Figures standing for numbers are called significant figures. They are 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Significant digit is found in 1876 in Elementary arithmetic, with brief notices of its history by Robert Potts: "7. If 5^-100000 be expressed as a decimal, how many ciphers will there be before the first significant digit of the decimal?" [University of Michigan Historical Math Collection].

Non-significant digit is found in January 1900 in Neal H. Ewing, "The Shakespeare Name," Catholic World: "Naught is the non-significant digit; though it means nothing, yet it counts for so much."

An article in The Mathematics Teacher in October 1939 explains that zero is sometimes a "significant figure."

SIMILAR. In 1557 Robert Recorde used like in the Whetstone of Witte: "When the sides of one plat forme, beareth like proportion together as the sides of any other flatte forme of the same kinde doeth, then are those formes called like flattes .. and their numbers, that declare their quantities, in like sorte are named like flattes" (OED2).

In the manuscript of his Characteristica Geometrica which was not published by him, Leibniz wrote "similitudinem ita notabimus: a ~ b."

In 1660 Isaac Barrow used like in his Euclid: "If in a triangle FBE there be drawn AC a parallel to one side FE, the triangle ABC shall be like to the whole FBE (OED2).

In English, similar triangles is found in 1704 in Lexicon technicum: "Similar Triangles are such as have all their three Angles respectively equal to one another" (OED2).

SIMPLE CLOSED CURVE occurs in 1873 in "On Listing's Theorem" by Arthur Cayley in the Messenger of Mathematics [University of Michigan Historical Math Collection].

SIMPLEX. William Kingdon Clifford (1848-1879) used the term prime confine in "Problem in Probability," Educational Times, Jan. 1886:

Now consider the analogous case in geometry of n dimensions. Corresponding to a closed area and a closed volume we have something which I shall call a confine. Corresponding to a triangle and to a tetrahedron there is a confine with n + 1 corners or vertices which I shall call a prime confine as being the simplest form of confine.

SIMPLEX METHOD is found in Robert Dorfman, "Application of the simplex method to a game theory problem," Activity Analysis of Production and Allocation, Chap. XXII, 348-358 (1951).

Simplex approach is found in 1951 by George B. Dantzig (1914- ) in T. C. Koopman's Activity Analysis of Production and Allocation xxi. 339: "The general nature of the 'simplex' approach (as the method discussed here is known)" (OED2).

SIMPLY ORDERED SET was defined by Cantor in Mathematische Annalen, vol. 46, page 496.

SIMPSON'S RULE is found in 1875 in An elementary treatise on the integral calculus by Benjamin Williamson (1827-1916): "This and the preceding are commonly called 'Simpson's rules' for calculating areas; they were however previously noticed by Newton" (OED2).

SIMSON LINE. The theorem was attributed to Robert Simson (1687-1768) by François Joseph Servois (1768-1847) in the Gergonne's Journal, according to Jean-Victor Poncelet in Traité des propriétés projectives des figures. The line does not appear in Simson's work and is apparently due to William Wallace. [The University of St. Andrews website]

SIMULTANEOUS EQUATIONS occurs in 1842 in Colenso, Elem. Algebra (ed. 3): "Equations of this kind, ... to be satisfied by the same pair or pairs of values of x and y, are called simultaneous equations" (OED2).

Simultaneous equations also appears in 1842 in G. Peacock, Treat. Algebra: "Such pairs or sets of equations in which the same unknown symbols appear, which are assumed to possess the same values throughout, are called simultaneous equations" (OED2).

SINE. Aryabhata the Elder (476-550) used the word jya for sine in Aryabhatiya, which was finished in 499.

According to Cajori (1906), the Latin term sinus was introduced in a translation of the astronomy of Al Battani by Plato of Tivoli (or Plato Tiburtinus).

According to some sources, sinus first appears in Latin in a translation of the Algebra of al-Khowarizmi by Gherard of Cremona (1114-1187). For example, Eves (page 177) writes:

The origin of the word sine is curious. Aryabhata called in ardha-jya ("half-chord") and also jya-ardha ("chord-half"), and then abbreviated the term by simply using jya ("chord"). From jya the Arabs phonetically derived jiba, which, following Arabian practice of omitting vowels, was written as jb. Now jiba, aside from its technical significance, is a meaningless word in Arabic. Later writers, coming across jb as an abbreviation for the meaningless jiba, substituted jaib instead, which contains the same letters and is a good Arabic word meaning "cove" or "bay." Still later, Gherardo of Cremona (ca. 1150), when he made his translations from the Arabic, replaced the Arabian jaib by its Latin equivalent, sinus, whence came our present word sine.

However, Boyer (page 278) places the first appearance of sinus in a translation of 1145. He writes:

It was Robert of Chester's translation from the Arabic that resulted in our word "sine." The Hindus had given the name jiva to the half chord in trigonometry, and the Arabs had taken this over as jiba. In the Arabic language there is also a word jaib meaning "bay" or "inlet." When Robert of Chester came to translate the technical word jiba, he seems to have confused this with the word jaib (perhaps because vowels were omitted); hence he used the word sinus, the Latin word for "bay" or "inlet." Sometimes the more specific phrase sinus rectus, or "vertical sine," was used; hence the phrase sinus versus, or our "versed sine," was applied to the "sagitta," or the "sine turned on its side."

Smith (vol. 1, page 202) writes that the Latin sinus "was probably first used in Robert of Chester's revision of the tables of al-Khowarizmi."

Fibonacci used the term sinus rectus arcus.

Regiomontanus (1436-1476) used sinus, sinus rectus, and sinus versus in De triangulis omnimodis (On triangles of all kinds; Nuremberg, 1533) [James A. Landau].

Copernicus and Rheticus did not use the term sine (DSB).

The earliest known use of sine in English is by Thomas Fale in 1593:

This Table of Sines may seem obscure and hard to those who are not acquainted with Sinicall computation.

The citation is above is from Horologiographia. The art of dialling: teaching an easie and perfect way to make all kinds of dials vpon any plaine plat howsoeuer placec: With the drawing of the twelue signes, and houres vnequall in them all... At London, Printed by Thomas Orwin, dwelling in Pater noster-Row ouer against the signe of the Checker, 1593, by Thomas Fale.

The term SINGLE-VALUED FUNCTION (meaning analytic function) was used by Yulian-Karl Vasilievich Sokhotsky (1842-1927).

The term SINGULAR INTEGRAL is due to Lagrange (Kline, page 532).

The term is found in 1831 in Elements of the Integral Calculus (1839) by J. R. Young:

We see, therefore, that it is possible for a differential equation to have other integrals besides the complete primitive, but derivable from it by substituting in it, for the arbitrary constant c, each of its values given in terms of x and y by the equation (5). Such integrals are called singular integrals, or singular solutions of the proposed differential equation.

SINGULAR MATRIX. Singular matrix and non-singular matrix occur in 1907 in Introduction to Higher Algebra by Maxime Bôcher: "Definition 2. A square matrix is said to be singular if its determinant is zero."

SINGULAR POINT appears in a paper by George Green published in 1828. The paper also contains the synonymous phrase "singular value" [James A. Landau].

Singular point appears in 1836 in the second edition of Elements of the Differential Calculus by John Radford Young. According to James A. Landau, who supplied this citation, it is not clear what the author meant by the term. Landau writes, "Judging by the contents of Chapter IV, to the author 'singular point' was the name of the category to which 'multiple points,' 'cusps,' and 'points of inflexion' belong."

In An Elementary Treatise on Curves, Functions and Forces (1846), Benjamin Peirce writes, "Those points of a curve, which present any peculiarity as to curvature or discontinuity, are called singular points."

SIZE (of a critical region) is found in 1933 in J. Neyman and E. S. Pearson, "On the Problems of the Most Efficient Tests of Statistical Hypotheses," Philosophical Transactions of the Royal Society of London, Ser. A (1933), 289-337 (David (2001)).

SKEW DISTRIBUTION appears in 1895 in a paper by Karl Pearson [James A. Landau].

SKEW SYMMETRIC MATRIX. Skew symmetric determinant appears in 1849 in Arthur Cayley, Jrnl. für die reine und angewandte Math. XXXVIII. 93: "Ces déterminants peuvent être nommés ‘gauches et symmétriques’" (OED2).

Skew symmetric determinant appears in 1885 in Modern Higher Algebra by George Salmon: "A skew symmetric determinant is one in which each constituent is equal to its conjugate with its sign changed."

Skew symmetric matrix appears in "Linear Algebras," Leonard Eugene Dickson, Transactions of the American Mathematical Society, Vol. 13, No. 1. (Jan., 1912).

SKEWES NUMBER appears in 1949 in Kasner & Newman, Mathematics and the Imagination: "A veritable giant is Skewes' number, even bigger than a googolplex" (OED2).

SLIDE RULE. In 1630, the terms Grammelogia and mathematical ring were used for a new device which, unlike Gunter's scale, had moving parts.

In 1632, the terms circles of proportion and horizontal instrument were used to describe Oughtred's device, in a 1632 publication, Circles of Proportion.

Slide rule appears in the Diary of Samuel Pepys (1633-1703) in April 1663: "I walked to Greenwich, studying the slide rule for measuring of timber." However, the device referred to may not have been a slide rule in the modern sense.

Slide rule appears in 1838 in Civil Eng. & Arch. Jrnl.: "To assist in facilitating the use of the slide rule among working mechanics" (OED2).

Amédée Mannheim (1831-1906) designed (c. 1850) the Mannheim Slide Rule.

Sliding-rule and sliding-scale appear in 1857 in Mathematical Dictionary and Cyclopaedia of Mathematical Science, defined in the modern sense.

Slide rule appears in 1876 in Handbk. Scientif. Appar.: "The slide rule,--an apparatus for effecting multiplications and divisions by means of a logarithmic scale" (OED2).

SLOPE is found in 1857 in Mathematical Dictionary and Cyclopedia of Mathematical Science:

SLOPE. Oblique direction. The slope of a plane is its inclination to the horizon. This slope is generally given by its tangent. Thus, the slope, 1/2, is equal to an angle whose tangent is 1/2; or, we generally say, the slope is 1 upon 2; that is, we rise, in ascending such a plane, a vertical distance of 1, in passing over a horizontal distance of 2. The slope of a curved surface, at any point, is the slope of a plane, tangent to the surface at that point.

In 1924 Analytic Geometry by Arthur M. Harding and George W. Mullins has: "If the line is parallel to the y axis, the slope is infinite." Modern textbooks say such a line has undefined slope.

For information on the use of m and other symbols for slope, see the entry on the math symbols page.

SLOPE-INTERCEPT FORM is found in 1904 in Elements of the Differential and Integral Calculus by William Anthony Granville [James A. Landau].

In Webster's New International Dictionary (1909), the term is slope form.

The term SOCIAL MATHEMATICS was used by Condorcet (1743-1794) and may have been coined by him.

SOLID GEOMETRY appears in 1733 in the title Elements of Solid Geometry by H. Gore (OED2).

SOLID OF REVOLUTION is found in English in 1816 in the translation of Lacroix's Differential and Integral Calculus: "To find the differentials of the volumes and curve surfaces of solids of revolution" (OED2).

SOLIDUS (the diagonal fraction bar). Arthur Cayley (1821-1895) wrote to Stokes, "I think the 'solidus' looks very well indeed...; it would give you a strong claim to be President of a Society for the Prevention of Cruelty to Printers" (Cajori vol. 2, page 313).

The word solidus appears in this sense in the Century Dictionary of 1891.

SOLUBLE (referring to groups). Ferdinand Georg Frobenius (1849-1917) wrote in a paper of 1893:

Jede Gruppe, deren Ordnung eine Potenz einer Primzahl ist, ist nach einem Satze von Sylow die Gruppe einer durch Wurzelausdrücke auflösbaren Gleichung oder, wie ich mich kurz ausdrücken will, einer auflösbare Gruppe. [Every group of prime-power order is, by a theorem of Sylow, the group of an equation which is soluble by radicals or, as I will allow myself to abbreviate, a soluble group.]

Peter Neumann believes this is likely to be the passage that introduced the term "auflösbar" ["soluble"] as an adjective applicable to groups into mathematical language.

SOLUTION SET appears in 1959 in Fund. Math. by Allendoerfer and Oakley: Given a universal set X and an equation F(x) = G(x) involving x, the set {x|F(x) = G(x)} is called the solution set of the given equation" (OED2).

The term may occur in found in Imsik Hong, "On the null-set of a solution for the equation $\Delta u+k^2u=0$," Kodai Math. Semin. Rep. (1955).

SOUSLIN SET is defined in Nicolas Bourbaki, Topologie Generale [Stacy Langton].

The term SPECIALLY MULTIPLICATIVE FUNCTION was coined by D. H. Lehmer (McCarthy, page 65).

SPECTRUM (in operator theory). The OED's earliest quotation illustrating the mathematical use of "spectrum" is from P. R. Halmos Finite Dimensional Vector Spaces (1948, ii. 79): "The set of n proper values [eigenvalues] of A, with multiplicities properly counted, is the spectrum of A." However the usage can be traced back to "Spektrum" in Hilbert's work on integral equations in 1904-10 and the elaboration of operator theory in the 1920's in works like von Neumann's "Allgemeine Eigenwerttheorie Hermitische Funktionaloperatoren" Math. Ann. 102 (1929) 49-131. M. H. Stone's Linear Transformations in Hilbert Space used the English word in 1932. The term SPECTRAL THEORY came into use in the early 1930's a few years after its German equivalent. (See also Eigenvalue and stationary stochastic process.) [John Aldrich]

SPECTRUM and SPECTRAL DENSITY (in generalised harmonic analysis and stochastic processes). The "spectrum" of an irregular motion appears in N. Wiener's "The Harmonic Analysis of Irregular Motion (Second Paper)" J. Math. and Phys. 5 (1926) 158-189. One of Wiener's objectives was a theory which would include "an adequate mathematical account of such continuous spectra as that of white light." (Wiener Proc. London Math. Soc. 27 (1928)) The term "power-spectrum" is also in the 1926 paper. The spectrum and spectral density function were important in the probabilistic theory of Khintchine (1934) and Wold (1938) but the functions were not given names. The names appear in J. L. Doob's "The Elementary Gaussian Processes" Annals of Mathematical Statistics, 15, (1944), 229-282. Around 1940 it became evident that the spectral theory of time series analysis was related to the spectral theory of operators. (See also the previous entry and stationary stochastic process). [John Aldrich]

SPHERICAL CONCHOID was coined by Herschel.

SPHERICAL GEOMETRY appears in 1728 in Chambers' Cyclopedia (OED2).

The words spherical geometry and versed sine were used by Edgar Allan Poe in his short story The Unparalleled Adventure Of One Hans Pfaall.

SPHERICAL HARMONICS. A. H. Resal used the term fonctions spheriques (Todhunter, 1873) [Chris Linton].

Spherical harmonics was used in 1867 by William Thomson (1824-1907) and Peter Guthrie Tait (1831-1901) in Nat. Philos.: "General expressions for complete spherical harmonics of all orders" (OED2).

SPHERICAL TRIANGLE Menelaus of Alexandria (fl. A. D. 100) used the term tripleuron in his Sphaerica, according to Pappus. According to the DSB, "this is the earliest known mention of a spherical triangle."

The OED2 shows a use of spherical triangle in English in 1585.

In a letter to L. H. Girardin dated March 18, 1814, Thomas Jefferson (President of the United States) wrote, "According to your request of the other day, I send you my formula and explanation of Lord Napier's theorem, for the solution of right-angled spherical triangles."

SPHERICAL TRIGONOMETRY is found in the title Trigonometria sphaericorum logarithmica (1651) by Nicolaus Mercator (1620-1687).

The term is found in English in a letter by John Collins to the Governors of Christ's Hospital written on May 16, 1682, in the phrase "plaine & spherick Trigonometry, whereby Navigation is performed" [James A. Landau].

In a letter dated Oct. 8, 1809, Thomas Jefferson wrote, referring to Benjamin Banneker, "We know he had spherical trigonometry enough to make almanacs, but not without the suspicion of aid from Ellicot, who was his neighbor and friend, and never missed an opportunity of puffing him."

SPINOR appears in 1931 in Physical Review. The citation refers to spinor analysis developed by B. Van der Waerden (OED2).

SPIRAL OF ARCHIMEDES appears in 1836 in the second edition of Elements of the Differential Calculus by John Radford Young [James A. Landau].

SPLINE CURVE is found in 1946 in I. J. Schoenberg, Q. Appl. Math. IV. 48: "For k = 4 they represent approximately the curves drawn by means of a spline and for this reason we propose to call them spline curves of order k (OED2).

The term SPORADIC GROUP was coined by William Burnside (1852-1927) in the second edition of his Theory of Groups of Finite Order, published in 1911 [John McKay].

SPURIOUS CORRELATION. The term was introduced by Karl Pearson in "On a Form of Spurious Correlation Which May Arise When Indices Are Used in the Measurement of Organs," Proc. Royal Society, 60, (1897), 489-498. Pearson showed that correlation between indices u (= x/z) and v (= x/z) was a misleading guide to correlation between x and y. His illustration is

A quantity of bones are taken from an ossuarium, and are put together in groups which are asserted to be those of individual skeletons. To test this a biologist takes the triplet femur, tibia, humerus, and seeks the correlation between the indices femur/humerus and tibia/humerus. He might reasonably conclude that this correlation marked organic relationship, and believe that the bones had really been put together substantially in their individual grouping. As a matter of fact ... there would be ... a correlation of about 0.4 to 0.5 between these indices had the bones been sorted absolutely at random.

The term has been applied to other correlation scenarios with potential for misleading inferences. In Student's "The Elimination of Spurious Correlation due to Position in Time or Space" (Biometrika, 10, (1914), 179-180) the source of the spurious correlation is the common trends in the series. In H. A. Simon's "Spurious Correlation: A Causal Interpretation," Journal of the American Statistical Association, 49, (1954), pp. 467-479 the source of the spurious correlation is a common cause acting on the variables. In the recent spurious regression literature in time series econometrics (Granger & Newbold, Journal of Econometrics, 1974) the misleading inference comes about through applying the correlation theory for stationary series to non-stationary series. The dangers of doing this were pointed out by G. U. Yule in his 1926 "Why Do We Sometimes Get Nonsense Correlations between Time-series? A Study in Sampling and the Nature of Time-series," Journal of the Royal Statistical Society, 89, 1-69. (Based on Aldrich 1995)

SQUARE MATRIX was used by Arthur Cayley in 1858 in Collected Math. Papers (1889): "The term matrix might be used in a more general sense, but in the present memoir I consider only square or rectangular matrices" (OED2).

The term STANDARD DEVIATION was introduced by Karl Pearson (1857-1936) in 1893, "although the idea was by then nearly a century old" (Abbott; Stigler, page 328). According to the DSB:

The term "standard deviation" was introduced in a lecture of 31 January, 1893, as a convenient substitute for the cumbersome "root mean square error" and the older expressions "error of mean square" and "mean error."

The OED2 shows a use of standard deviation in 1894 by Pearson in "Contributions to the Mathematical Theory of Evolution, Philosophical Transactions of the Royal Society of London, Ser. A. 185, 71-110: "Then will be termed its standard-deviation (error of mean square)."

STANDARD ERROR is found in 1897 in G. U. Yule, "On the Theory of Correlation," Journal of the Royal Statistical Society, 60, 812-854: "We see that ₁[sqrt](1 - r²) is the standard error made in estimating x" (OED2).

STANDARD POSITION is dated 1950 in MWCD10.

STANDARD SCORE. In 1921 Univ. Illin. Bur. Educ. Res. Bull. has: "Provision is made for comparing a pupil's achievement score..with the norm corresponding to his mental age by dividing his achievement age by the standard score for his mental age. This quotient is called the Achievement Quotient" (OED2).

Standard score is dated 1928 in MWCD10.

STANINE is dated 1944 in MWCD10.

The earliest citation in the OED2 is from the Baltimore Sun, Oct. 1, 1945, "The result .. was a 'stanine' rating (stanine being an invented word, from 'standard of nine')."

Stanines were first used to describe an examinee's performance on a battery of tests constructed for the U. S. Army Air Force during World War II.

The term STAR PRIME was coined in 1988 by Richard L. Francis (Schwartzman, p. 206).

STATIONARY STOCHASTIC PROCESS appears in the title of A Khintchine's "Korrelationstheorie der Stationären Stochastischen Prozesse," Math. Ann. 109, 604.

H. Wold translated it as "stationary random process" (A Study in the Analysis of Stationary Time Series (1938)).

The phrase "stationary stochastic process" appears in J. L. Doob's "What is a Stochastic Process?" American Mathematical Monthly, 49, (1942), 648-653.

An older term was "fonction éventuelle homogène," which appears in E. Slutsky's "Sur les Fonctions Éventuelles Continues, Intégrables et Dérivables dans la Sens Stochastique," Comptes Rendues, 187, (1928), 878 [John Aldrich].

STATISTIC (as opposed to parameter) is found in R. A. Fisher, "On the Mathematical Foundations of Theoretical Statistics," Philosophical Transactions of the Royal Society of London, Ser. A., 222, 309-368: "These involve the choice of methods of calculating from a sample statistical derivates, or as we shall call them statistics, which are designed to estimate the values of the parameters of the hypothetical population" (OED2).

This term was introduced in 1922 by Fisher, according to Tankard (p. 112).

The term statistic was not well-received initially. Arne Fisher (no relation) asked Fisher, "Where ... did you get that atrocity, a statistic? (letter (p. 312) in J. H. Bennet Statistical Inference and Analysis: Selected Correspondence of R. A. Fisher (1990).) Karl Pearson objected, "Are we also to introduce the words a mathematic, a physic, an electric etc., for parameters or constants of other branches of science?" (p. 49n of Biometrika, 28, 34-59 1936). [These two quotations were provided by John Aldrich.]

STATISTICS originally referred to political science and it is difficult to determine when the word was first used in a purely mathematical sense. The earliest citation of the word statistics in the OED2 is in 1770 in W. Hooper's translation of Bielfield's Elementary Universal Education: "The science, that is called statistics, teaches us what is the political arrangement of all the modern states of the known world." However, there are earlier citations for statistical and Latin and German forms of statistic, all used in a political sense.

In Webster's dictionary of 1828 the definition of statistics is: "A collection of facts respecting the state of society, the condition of the people in a nation or country, their health, longevity, domestic economy, arts, property and political strength, the state of the country, &c."

STEP FUNCTION is dated ca. 1929 in MWCD10.

STEREOGRAPHIC. According to Schwartzman (p. 207), "the term seems to have been used first by the Belgian Jesuit François Aguillon (1566-1617), although the concept was already known to the ancient Greeks."

In Flattening the Earth: Two Thousand Years of Map Projections, John P. Snyder attributes the term to d'Aguillon in 1613 [John W. Dawson, Jr.].

STIELTJES INTEGRAL is found in Henri Lebesgue, "Sur l'intégrale de Stieltjes et sur les opérations linéaires," Comptes Rendus Acad. Sci. Paris 150 (1910) [James A. Landau].

The terms STIRLING NUMBERS OF THE FIRST and SECOND KIND were coined by Niels Nielsen (1865-1931), who wrote in German "Stirlingschen Zahlen erster Art" [Stirling numbers of the first kind] and "Stirlingschen Zahlen zweiter Art" [Stirling numbers of the second kind]. Nielsen's masterpiece, "Handbuch der Theorie der Gammafunktion" [B. G. Teubner, Leipzig, 1906], had a great influence, and the terms progressively found their acceptance (Julio González Cabillón).

John Conway believes the newer terms Stirling cycle and Stirling (sub)set numbers were introduced by R. L. Graham, D. E. Knuth, and O. Patshnik in Concrete Mathematics (Addison Wesley, 1989 & often reprinted).

STIRLING'S FORMULA. Lacroix used Théorème de Stirling in Traité élémentaire de calcul différentiel et de calcul intégral (1797-1800).

Stirling's approximation appears in 1938 in Biometrika (OED2).

STOCHASTIC is found in English as early as 1662 with the obsolete meaning "pertaining to conjecture."

In its modern sense, the term was used in 1917 by Ladislaus Josephowitsch Bortkiewicz (1868-1931) in Die Iterationem 3: "Die an der Wahrscheinlichkeitstheorie orientierte, somit auf 'das Gesetz der Grossen Zahlen' sich gründende Betrachtng empirischer Vielheiten mö ge als Stochastik ... bezeichnet werden" (OED2).

Stochastic process is found in A. N. Kolmogorov, "Sulla forma generale di un prozesso stocastico omogeneo," Rend. Accad. Lincei Cl. Sci. Fis. Mat. 15 (1) page 805 (1932) [James A. Landau].

Stochastic process is also found in A. Khintchine "Korrelationstheorie der stationäre stochastischen Prozesse," Math. Ann. 109 (1934) [James A. Landau].

Stochastic process occurs in English in "Stochastic processes and statistics," Proc. Natl. Acad. Sci. USA 20 (1934).

STOKES'S THEOREM. According to Finney and Thomas (page 987), Stokes learned of the theorem from Lord Kelvin in 1850 and "a few years later, thinking it would make a good examination question, put it on the Smith Prize examination. It has been known as Stokes's theorem ever since."

Stokes' theorem is found in 1893 in J. J. Thomsom, Notes Recent Res. Electr. & Magnetism (OED2).

STRAIGHT ANGLE appears in English in 1889 in Dupuis, Elem. Synth. Geom.: "One-half of a circumangle is a straight angle, and one-fourth of a circumangle is a right angle" (OED2).

There are earlier citations in the OED2 for the term with the obsolete meaning of "a right angle."

The term STRANGE ATTRACTOR was coined by David Ruelle and Floris Takens in their classic paper "On the Nature of Turbulence" [Communications in Mathematical Physics, vol. 20, pp. 167-192, 1971], in which they describe the complex geometric structure of an attractor during a study of models for turbulence in fluid flow.

STRATIFIED SAMPLING occurs in J. Neyman, "On the two different aspects of the representative method; the method of stratified sampling and the method of purposive selection," J. R. Satatist. Soc 97 (1934) [James A. Landau].

STRONG LAW OF LARGE NUMBERS is found in A. N. Kolmogorov, "Sur la loi forte des grandes nombres," Comptes Rendus de l'Acade/mie des Sciences, Paris 191 page 910 (1930) [James A. Landau].

STRONG PSEUDOPRIME. According to Prime Numbers: A Computational Perspective by Carl Pomerance and Richard Crandall (page 124), "J. Selfridge proposed using Theorem 3.4.1 as a pseudoprime test in the early 1970s, and it was he who coined the term 'strong pseudoprime'" [Paul Pollack].

Strong pseudoprime is found in Pomerance, Carl; Selfridge, J.L.; Wagstaff, Samuel S. Jr. "The pseudoprimes to 25 x 10⁹," Math. Comput. 35, 1003-1026 (1980).

STROPHOID appears in 1837 in Enrico Montucci, "Delle proprietà della strefoide, curva algebrica del terzo grado recentemente scoperta ed esaminata" ("On the property of the strophoid, an algebraic curve of the third degree recently discovered and examined"), Memoria letta nell'Accademia dei Fisiocratici ... con una appendice del Venturoli, Siena, G. Mucci, 1837 [Dic Sonneveld].

Strophoid was coined by Montucci in 1846, according to Smith (vol. 2, page 330).

The term STRUCTURE for isomorphic relations seems to have first appeared in print in Bertrand Russell's Introduction to Mathematical Philosophy (1919). Russell probably had the term from Ludwig Wittgenstein, whose Tractatus logico-philosophicus (Logisch-philosophische Abhandlung, Vienna 1918, 4.1211 ff) was first published in 1921, and in 1922 in English. The first Structure in the modern sense -- as a tuple composed of sorts or carrier sets, relations, operations and distinguished elements -- was first used by David Hilbert in his Grundlagen der Geometrie (Göttingen 1899), there called a „Fachwerk oder Schema von Begriffen“ (p. 163, according to F. Kambartel Erfahrung und Struktur, Münster 1966). The concept of Structure developed via Rudolf Carnap's Der logische Aufbau der Welt (1928), the linguistic and French philosophical Structuralism, the Éléments de mathématique of the N. Bourbaki group (Paris, since 1939), to Category Theory of Samuel Eilenberg and Saunders Mac Lane (1945). [This entry was contributed by Wolfram Roisch.]

STUDENT'S t-DISTRIBUTION. "Student" was the pseudonym of William Sealy Gosset (1876-1937). Gosset once wrote to R. A. Fisher, "I am sending you a copy of Student's Tables as you are the only man that's ever likely to use them!" The letter appears in Letters from W. S. Gosset to R. A. Fisher, 1915-1936 (1970). Student's tables became very important in statistics but not in the form he first constructed them.

In his 1908 paper, "The Probable Error of a Mean," Biometrika 6, 1-25 Gosset introduced the statistic, z, for testing hypotheses on the mean of the normal distribution. Gosset used the divisor n, not the modern (n - 1), when he estimated and his z is proportional to t with t = z sqrt (n - 1). Fisher introduced the t form for it fitted in with his theory of degrees of freedom. Fisher's treatment of the distributions based on the normal distribution and the role of degrees of freedom was given in "On a Distribution Yielding the Error Functions of Several well Known Statistics," Proceedings of the International Congress of Mathematics, Toronto, 2, 805-813. The t symbol appears in this paper but although the paper was presented in 1924, it was not published until 1928 (Tankard, page 103; David, 1995). According to the OED2, the letter t was chosen arbitrarily. A new symbol suited Fisher for he was already using z for a statistic of his own (see entry for F).

Student's distribution (without "t") appears in 1925 in R. A. Fisher, "Applications of 'Student's' Distribution," Metron 5, 90-104 and in Statistical Methods for Research Workers (1925). The book made Student's distribution famous; it presented new uses for the tables and made the tables generally available.

"Student's" t-distribution appears in 1929 in Nature (OED2).

t-distribution appears (without Student) in A. T. McKay, "Distribution of the coefficient of variation and the extended 't' distribution," J. Roy. Stat. Soc., n. Ser. 95 (1932).

t-test is found in 1932 in R. A. Fisher, Statistical Methods for Research Workers: "The validity of the t-test, as a test of this hypothesis, is therefore absolute" (OED2).

Eisenhart (1979) is the best reference for the evolution of t, although Tankard and Hald also discuss it.

[This entry was largely contributed by John Aldrich.]

STUDENTIZATION. According to Hald (p. 669), William Sealy Gossett (1876-1937) used the term Studentization in a letter to E. S. Pearson of Jan. 29, 1932.

Studentized D² statistic is found in R. C. Bose and S. N. Roy, "The exact distribution of the Studentized D² statistic," Sankhya 3 pt. 4 (1935) [James A. Landau].

STURM'S THEOREM appears in 1836 in the title Du Theoreme de M. Sturm, et de ses Applications Numeriques by M. E. Midy [James A. Landau].

Sturm's theorem appears in English in 1841 in the title Mathematical Dissertations, for the use of students in the modern analysis; with improvements in the practice of Sturm's Theorem, in the theory of curvature, and in the summation of infinite series by J. R. Young [James A. Landau].

SUBFACTORIAL was introduced in 1878 by W. Allen Whitworth in Messenger of Mathematics (Cajori vol. 2, page 77).

SUBFIELD is found in "On the Base of a Relative Number-Field, with an Application to the Composition of Fields," G. E. Wahlin, Transactions of the American Mathematical Society, Vol. 11, No. 4. (Oct., 1910).

SUBGROUP. Felix Klein used the term untergruppe.

Subgroup appears in 1881 in Arthur Cayley, "On the Schwarzian Derivative, and the Polyhedral Functions," Transactions of the Cambridge Philosophical Society: "But there is no sub-group of an order divisible by 5; and hence, these two transformations being identified with the two substitutions, the other transformations correspond each of them to a determinate substitution" [University of Michigan Historical Math Collection].

SUBRING is found in English in 1937 in the phrase invariant subring in Modern Higher Algebra (1938) by A. A. Albert (OED2).

SUBSET. Cantor used the word subset (in the sense that "proper subset" is now used) in "Ein Beitrag zur Mannigfaltigkeitslehre," Journal für die reine und angewandte Mathematik 84 (1878).

Subset occurs in English in "A Simple Proof of the Fundamental Cauchy-Goursat Theorem," Eliakim Hastings Moore, Transactions of the American Mathematical Society, Vol. 1, No. 4. (Oct., 1900).

SUBTRACT. When Fibonacci (1201) wishes to say "I subtract," he uses some of the various words meaning "I take": tollo, aufero, or accipio. Instead of saying "to subtract" he says "to extract."

In English, Chaucer used abate around 1391 in Treatise on the Astrolabe: "Abate thanne thees degrees And minutes owt of 90" (OED2).

In a manuscript written by Christian of Prag (c. 1400), the word "subtraction" is at first limited to cases in which there is no "borrowing." Cases in which "borrowing" occurs he puts under the title cautela (caution), and gives this caption the same prominence as subtractio.

In Practica (1539) Cardano used detrahere (to draw or take from).

In 1542 in the Ground of Artes Robert Recorde used rebate: "Than do I rebate 6 out of 8, & there resteth 2."

In 1551 in Pathway to Knowledge Recorde used abate: "Introd., And if you abate euen portions from things that are equal, those partes that remain shall be equall also" (OED2).

Digges (1572) writes "to subduce or substray any sume, is wittily to pull a lesse fro a bigger number."

Schoner, in his notes on Ramus (1586 ed., p. 8), uses both subduco and tollo for "I subtract."

In his arithmetic, Boethius uses subtrahere, but in geometry attributed to him he prefers subducere.

The first citation for subtract in the OED2 is in 1557 by Robert Recorde in The whetstone of witte: "Wherfore I subtract 16. out of 18."

Hylles (1592) used "abate," "subtact," "deduct," and "take away" (Smith vol. 2, pages 94-95).

From Smith (vol. 2, page 95):

The word "subtract" has itself had an interesting history. The Latin sub appears in French as sub, soub, sou, and sous, subtrahere becoming soustraire and subtractio becoming soustraction. Partly because of this French usage, and partly no doubt for euphony, as in the case of "abstract," there crept into the Latin works of the Middle Ages, and particularly into the books printed in Paris early in the 16th century, the form substractio. From France the usage spread to Holland and England, and form each of these countries it came to America. Until the beginning of the 19th century "substract" was a common form in England and America, and among those brought up in somewhat illiterate surroundings it is still to be found. The incorrect form was never popular in Germany, probably because of the Teutonic exclusion of international terms.

SUBTRACTION. Fibonacci (1201) used extractio.

Tonstall (1522) devoted 15 pages to Subductio. He wrote, "Hanc autem eandem, uel deductionem uel subtractionem appellare Latine licet" (1538 ed., p. 23; 1522 ed., fol. E 2, r).

Gemma Frisius (1540) has a chapter De Subductione siue Subtractione.

Clavius (1585 ed., p. 26) says "Subtractio est ... subductio."