# Earliest Known Uses of Some of the Words of Mathematics (S)

Last revision: Oct. 1, 1999

ST. ANDREW'S CROSS is the term used by Florian Cajori in History of Mathematical Notations for the multiplication symbol X.

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).

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."

SAMPLE SPACE appears in W. Feller, "Note on regions similar to the sample space," Statist. Res. Mem., Univ. London 2, 117-125 (1938).

The term may have been used earlier by Richard von Mises (1883-1953).

SCALAR. See vector.

SCALAR PRODUCT. See vector product.

SCALENE. In Sir Henry Billingsley's 1570 translation of Euclid's Elements scalenum is used as a noun: "Scalenum is a triangle, whose three sides are all unequall."

In 1642 scalene is found in a rare use as a noun, referring to a scalene triangle in Song of Soul by Henry More: "But if 't consist of points: then a Scalene I'll prove all one with an Isosceles."

The earliest use of scalene as an adjective is in 1684 in Angular Sections by John Wallis: "The Scalene Cone and Cylinder."

The earliest use of scalene as an adjective to describe a triangle is in 1734 in The Builder's Dictionary. (All citations are from the OED2.)

SCATTER DIAGRAM is found in 1925 in F. C. Mills, Statistical Methods X. 366: "The equation to a straight line, fitted by the method of least squares to the points on the scatter diagram, will express mathematically the average relationship between these two variables" (OED2).

Scattergram is found in 1938 in A. E. Waugh, Elem. Statistical Method: "This is the method of plotting the data on a scatter diagram, or scattergram, in order that one may see the relationship" (OED2).

SCIENTIFIC NOTATION. In 1895 in Computation Rules and Logarithms Silas W. Holman referred to the notation as "the notation by powers of ten." In the preface, which is dated August 1895, he wrote: "The following pages contain ... an explanation of the use of the notation by powers of ten ... the notation by powers of 10, as in the explanation here given. It seems unfortunate that this simple notation, so useful in computation and so great an aid in the explanation of numerical relations, is not universally incorporated into arithmetical instruction." [James A. Landau]

In A Scrap-Book of Elementary Mathematics (1908) by William F. White, the notation is called the index notation.

Scientific notation appears in Webster's Second New International Dictionary (1934), which says that numbers in this format are sometimes called condensed numbers.

Other terms are exponential notation and standard notation.

SECANT (in trigonometry) was introduced by Thomas Fincke (1561-1656) in his Thomae Finkii Flenspurgensis Geometriae rotundi libri XIIII, Basileae: Per Sebastianum Henricpetri, 1583. (His name is also spelled Finke, Finck, Fink, and Finchius.) Fincke wrote secans in Latin.

Vieta (1593) did not approve of the term secant, believing it could be confused with the geometry term. He used Transsinuosa instead (Smith vol. 2, page 622).

SELF-CONJUGATE. Kramer (p. 388) says Galois used this term, referring to a normal subgroup.

The term SEMI-CUBICAL PARABOLA was coined by John Wallis (Cajori 1919, page 181).

SEMIGROUP appears in English in 1904 in the Bulletin of the American Mathematical Society (OED2).

SEMI-INVARIANT appears in R. Frisch, "Sur les semi-invariants et moments employés dans l'étude des distributions statistiques," Oslo, Skrifter af det Norske Videnskaps Academie, II, Hist.-Folos. Klasse, no. 3 (1926) [James A. Landau].

SENTENTIAL CALCULUS is dated 1937 in MWCD10.

SEPARABLE appears in 1831 in Elements of the Integral Calculus (1839) by J. R. Young: "We shall first consider the general form X dy + Y dx = 0, which is the simplest for which the variables are separable: X being a function of x without y, and Y a function of y without x.

SEQUENCE. The OED2 shows a use by Sylvester in 1882 in the American Journal of Mathematics with the "rare" definition of a succession of natural numbers in order.

Sequence appears with its modern mathematical definition 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, Vol. 1, No. 3. (Jul., 1900). [The term may be considerably older.]

SERIES. According to Smith (vol. 2, page 481), "The early writers often used proportio to designate a series, and this usage is found as late as the 18th century."

John Collins (1624-1683) wrote to James Gregory on Feb. 2, 1668/1669, "...the Lord Brouncker asserts he can turne the square roote into an infinite Series" (DSB, article: "Newton").

Series is found in 1676 in Collins in Rigaurd Corr. Sci. Men (1841) II. 10: "The reducibility of Davenant’s problem to infinite series" (OED2).

According to Smith (vol. 2, page 497), "The change to the name 'series' seems to have been due to writers of the 17th century. ... Even as late as the 1693 edition of his algebra, however, Wallis used the expression 'infinite progression' for infinite series."

In the English translation of Wallis' algebra (translated by him and published in 1685), Wallis wrote:

Now (to return where we left off:) Those Approximations (in the Arithmetick of Infinites) above mentioned, (for the Circle or Ellipse, and the Hyperbola;) have given occasion to others (as is before intimated,) to make further inquiry into that subject; and seek out other the like Approximations, (or continual approaches) in other cases. Which are now wont to be called by the name of Infinite Series, or Converging Series, or other names of like import.
The SERPENTINE curve was named by Isaac Newton (1642-1727) in 1701, according to Schwartzman and the University of St. Andrews website.

The term SET first appears in Paradoxien des Unendlichen (Paradoxes of the Infinite), Hrsg. aus dem schriftlichen Nachlasse des Verfassers von Fr. Prihonsky, C. H. Reclam sen., xi, pp. 157, Leipzig, 1851. This small tract by Bernhard Bolzano (1781-1848) was published three years after his death by a student Bolzano had befriended (Burton, page 592).

Menge (set) is found in Geometrie der Lage (2nd ed., 1856) by Carl Georg Christian von Staudt: "Wenn man die Menge aller in einem und demselben reellen einfoermigen Gebilde enthaltenen reellen Elemente durch n + 1 bezeichnet und mit diesem Ausdrucke, welcher dieselbe Bedeutung auch in den acht folgenden Nummern hat, wie mit einer endlichen Zahl verfaehrt, so ..." [Ken Pledger].

Georg Cantor (1845-1918) did not define the concept of a set in his early works on set theory, according to Walter Purkert in Cantor's Philosophical Views.

Cantor's first definition of a set appears in an 1883 paper: "By a set I understand every multitude which can be conceived as an entity, that is every embodiment [Inbegriff] of defined elements which can be joined into an entirety by a rule." This quotation is taken from Über unendliche lineare Punctmannichfaltigkeiten, Mathematische Annalen, 21 (1883).

In 1895 Cantor used the word Menge in Beiträge zur Begründung der Transfiniten Mengenlehre, Mathematische Annalen, 46 (1895):

By a set we understand every collection [Zusammenfassung] M of defined, well-distinguished objects m of our intuition [Zusammenfassung] or our thinking (which are called the elements of M brought together to form an entirety.
This translation was taken from Cantor's Philosophical Views by Walter Purkett.

SET THEORY appears in Georg Cantor, "Sur divers théorèmes de lat théorie des ensembles de points situés dans un espace continu à n dimensions. Première communication." Acta Mathematica 2, pp. 409-414 (1883) [James A. Landau].

The term is also found in Ivar Bendixson, "Quelques théorèmes de la théorie des ensembles de points," Acta Mathematica 2, pp. 415-429 (1883) [James A. Landau].

In a letter to Mittag-Leffler, Cantor wrote on May 5, 1883, "Unfortunately, I am prevented by many circumstances from working regularly, and I would be fortunate to find, in you and your distinguished students, coworkers who probably will soon surpass me in 'set theory.'" This quotation, which is presumably a translation, was taken from Cantor's Continuum Problem by Gregory H. Moore.

Set theory is found in English in 1926 in Annals of Mathematics (2d ser.) XXVII. 487: "An important idea in set theory is that of relativity" (OED2 update).

SHORT DIVISION appears in the Century Dictionary (1889-97).

SIEVE OF ERATOSTHENES is found in English in 1803 in a translation of Bossut's Gen. Hist. Math.: "The famous sieve of Eratosthenes..affords an easy and commodious method of finding prime numbers" (OED2).

SIGN OF AGGREGATION is found in 1900 in Teaching of Elementary Mathematics by David Eugene Smith: "Signs of aggregation often trouble a pupil more than the value of the subject warrants. The fact is, in mathematics we never find any such complicated concatenations as often meet the student almost on the threshold of algebra."

SIGN TEST appears in W. MacStewart, "A note on the power of the sign test," Ann. Math. Statist. 12 (1941) [James A. Landau].

SIGNED NUMBER. Signed magnitude appears in 1873 in Proc. Lond. Math. Soc.: "A signed magnitude" (OED2).

Signed number appears in the title "The [Arithmetic] Operations on Signed Numbers" by Wilson L. Miser in Mathematics Magazine (1932).

SIGNIFICANCE TEST. Test of significance is found in 1907 in Biometrika V. 183: " Several other cases of probable error tests of significance deserve reconsideration" (OED2).

Testing the significance is found in "New tables for testing the significance of observations," Metron 5 (3) pp 105-108 (1925) [James A. Landau].

Test of significance is found in 1935 in Proc. Cambr. Philos. Soc. XXXI. 230: "The test of significance of a correlation coefficient has been shown..to be little vitiated by non-normality" (OED2).

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 1827 in Hutton, Course Math.: "The first nine are called Significant Figures, as distinguished from the cipher, which is of itself quite insignificant" (OED2).

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.
The term SIMILAR was used in Latin by Leibniz, but may be much older.

SIMPLE CLOSED CURVE occurs in "Theory on Plane Curves in Non-Metrical Analysis Situs," Oswald Veblen, Transactions of the American Mathematical Society, Vol. 6, No. 1. (Jan., 1905).

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 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.'

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. Non-singular matrix (and presumably singular matrix also) occurs in "Resolution into Involutory Substitutions of the Transformations of a Non-Singular Bilinear form into Itself," Dunham Jackson, Transactions of the American Mathematical Society, Vol. 10, No. 4. (Oct., 1909).

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

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."

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 1911 in T. Muir, Hist. Determinants: "Any skew determinant is expressible in terms of skew symmetric determinants and those of the original determinant which are not included in the latter" (OED2).

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 gogolplex" (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.
SLOPE-INTERCEPT FORM. In Webster's New International Dictionary (1909) and in A Brief Course in Advanced Algebra (1937), the term is slope form.

Slope-intercept form is dated ca. 1942 in MWCD10.

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).

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

The term SPECTRUM (the set of eigenvalues of a Hermitian operator) was coined by Hilbert (DSB).

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."

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].

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

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 the 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.'"

STANDARD ERROR is found in 1897 in G. U. Yule, Jrnl. R. Statist. Soc. 60: "We see that [sigma]1[sqrt](1 - r2) is the standard error made in estimating x" (OED2).

STANDARD POSITION is dated 1950 in MWCD10.

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

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.

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 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 was coined by E. Montucci in 1846, according to Schwartzman (page 209) and Smith (vol. 2, page 330). (However, Julio González Cabillón says that, in reading the article "La Strophoide" (1846), it does not seem that Montucci is introducing a new term.)

STUDENT'S t-DISTRIBUTION. The letter t for this distribution occurs in "New tables for testing the significance of observations," Metron 5 (3) (1925). This may be its first appearance in print [James A. Landau].

"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).

Student's distribution (without "t") appears in R. A. Fisher, "Applications of 'Student's' Distribution," Metron, 3, 90-114. This paper is dated both 1925 and 1926 in separate bibliographies [James A. Landau].

"Student" was the pseudonym of William Sealy Gossett (1876-1937).

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 [James A. Landau].

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 1853 in J. J. Sylvester, Phil. Trans. CXLIII. 483: "Reverting now to the simplified Sturmian residues, since..these differ from the unsimplified complete residues required by the Sturmian method only in the circumstance of their being divested of factors, which are necessarily..positive, these simplified Sturmians may of course be substituted for the complete Sturmians for the purposes of M. Sturm’s theorem" (OED2).

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 1887 in Amer. Jrnl. Math. IX. 51: "I use 'self-conjugate sub-group' in translating Klein’s ‘ausgezeichnete Untergruppe’ and Jordan’s ‘groupe permutable’" (OED2).

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

SUBSET occurs 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 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."

SUBTRAHEND is an abbreviation of the Latin numerus subtrahendus (number to be subtracted).

SUCCESSIVE INDUCTION. This term was suggested by Augustus De Morgan in his article "Induction (Mathematics)" in the Penny Cyclopedia of 1838. See also mathematical induction, induction, complete induction.

The phrase SUFFICIENT STATISTIC is found in 1922 in R. A. Fisher in Philosophical Transactions of the Royal Society:

In the case of the normal curve of distribution it is evident that the second moment is a sufficient statistic for estimating the standard deviation; in investigating a sufficient solution for grouped normal data, we are therefore in reality finding the optimum correction for grouping; the Sheppard correction having been proved only to satisfy the criterion of consistency.
SUM. Nicolas Chuquet used some in his Triparty en la Science des Nombres in 1484.

The term SUMMABLE (referring to a function that is Lebesgue integrable such that the value of the integral is finite) was introduced by Lebesgue (Klein, page 1045).

SUPPLEMENT. "Supplement of a parallelogram" appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements.

In 1704 Lexicon Technicum by John Harris has "supplement of an Ark."

In 1796 Hutton Math. Dict. has "The complement to 180° is usually called the supplement.

In 1798 Hutton in Course Math. has "supplemental arc" (one of two arcs which add to a semicircle) (OED2).

Supplement II to the 1801 Encyclopaedia Britannica has, "The supplement of 50° is 130°; as the complement of it is 40 °" (OED2).

In 1840, Lardner in Geometry vii writes, "If a quadrilateral figure be inscribed in a circle, its opposite angles will be supplemental" (OED2).

Supplementary angle is dated ca. 1924 in MWCD10.

SURD. According to Smith (vol. 2, page 252), al-Khowarizmi (c. 825) referred to rational and irrational numbers as 'audible' and 'inaudible', respectively.

This was translated as surdus ("deaf" or "mute") in Latin.

As far as is known, the first known European to adopt this terminology was Gherardo of Cremona (c. 1150).

Fibonacci (1202) adopted the same term to refer to a number that has no root, according to Smith.

Surd is found in English in Robert Recorde's The Pathwaie to Knowledge (1551): "Quantitees partly rationall, and partly surde" (OED2).

According to Smith (vol. 2, page 252), there has never been a general agreement on what constitutes a surd. It is admitted that a number like sqrt 2 is a surd, but there have been prominent writers who have not included sqrt 6, since it is equal to sqrt 2 X sqrt 3. Smith also called the word surd "unnecessary and ill-defined" in his Teaching of Elementary Mathematics (1900).

G. Chrystal in Algebra, 2nd ed. (1889) says that "...a surd number is the incommensurable root of a commensurable number," and says that sqrt e is not a surd, nor is sqrt (1 + sqrt 2).

SURJECTION appears in 1964 in Foundations of Algebraic Topology by W. J. Pervin (OED2).

SURJECTIVE appears in 1956 in C. Chevalley, Fund. Concepts Algebra: "A homomorphism which is injective is called a monomorphism; a homomorphism which is surjective is called an epimorphism" (OED2).

The term SURREAL NUMBER was introduced by Donald Ervin Knuth (1938- ) in 1972 or 1973, although the notion was previously invented by John Horton Conway (1937- ) in 1969.

The term SYLOW'S THEOREM is found in 1893 in Proceedings of the London Mathematical Society XXV 14 (OED2).

The term SYMMEDIAN was introduced in 1883 by Philbert Maurice d'Ocagne (1862-1938) [Clark Kimberling].

The term SYMMEDIAN POINT (point of concurrence of symmedians) was coined by the Robert Tucker (1832-1905) in the interest of uniformity and amity. The point had been called the Lemoine point in France and the Grebe point (after E. W. Grebe) in Germany (DSB, article "Lemoine").

The term SYMPLECTIC GROUP was proposed in 1939 by Herman Weyl in The Classical Groups. He wrote on page 165:

The name "complex group" formerly advocated by me in allusion to line complexes, as these are defined by the vanishing of antisymmetric bilinear forms, has become more and more embarrassing through collision with the word "complex" in the connotation of complex number. I therefore propose to replace it by the corresponding Greek adjective "symplectic." Dickson calls the group the "Abelian linear group" in homage to Abel who first studied it.
[This information was provided by William C. Waterhouse.]

SYNTHETIC DIVISION is found in 1857 in Mathematical Dictionary and Cyclopedia of Mathematical Science.

SYNTHETIC GEOMETRY appears in 1889 in the title Elementary Synthetic Geometry of the Point, Line and Circle in the Plane, by N. F. Dupuis (OED2). It also appears in the Century Dictionary (1889-97).

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