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

OBLIQUE ANGLE. In 1594 Thomas Blundevil referred to "oblique angled Triangles" in Exercises.

Oblique angle appears in English in 1695 in Geometry Epitomized by William Alingham: "An Oblique Angle, is either Acute or Obtuse" [OED].

OBTUSE ANGLE. See the entry RIGHT, OBTUSE and ACUTE ANGLES.

OBTUSE TRIANGLE. Amblygonium and obtuse angled triangle appear in 1570 Billingsley's translation of Euclid: "An ambligonium or an obtuse angled triangle" [OED].

Amblygone is found in English in 1623; amblygon is found in 1706.

Obtusangulous is found in English in 1680; obtusangular is found in 1706.

In 1828 Webster's dictionary has: "If one of the angles is obtuse, the triangle is called obtusangular or amblygonous."

Obtuse triangle is found in 1852 in The Lives of Winfield Scott and Andrew Jackson by Josh Tyler Headley: "Thus the American army stood like an obtuse triangle of which the British line formed the base." [Google print search]

OCTAGON. Octoagon is found in 1594 in A. I. Cameron, Warrender Papers (1932): "Table formed of thrie partis of octoagon." [OED]

ODD FUNCTION is found in 1819 in New Series of The Mathematical Repository by Thomas Leybourn [Google print search].

ODD NUMBER and EVEN NUMBER. The Pythagoreans knew of the distinction between odd and even numbers. The Pythagoreans used the term gnomon for the odd number.

A fragment of Philolaus (c. 425 B. C.) says that "numbers are of two special kinds, odd and even, with a third, even-odd, arising from a mixture of the two."

Euclid, Book 7, definition 6 is "An even number is that which is divisible into two parts."

Odd and even are found in English in various Middle Age documents including Art of Nombrynge (around 1430) "Compt the nombre of the figures, and wete yf it be ode or even" [OED].

In English, gonomon is found in 1660 in Stanley, Hist. Philos. (1701): "Odd Numbers they called Gnomons, because being added to Squares, they keep the same Figures; so Gnomons do in Geometry" [OED].

ODD PERMUTATION and EVEN PERMUTATION are found in "On the Relation between the Three-Parameter Groups of a Cubic Space Curve and a Quadric Surface," A. B. Coble, Transactions of the American Mathematical Society, Vol. 7, No. 1. (Jan., 1906).

Odd permutation is found in the fourth edition of Determinants (1906) by Laenas Gifford Weld, and may appear in earlier editions.

ODDS. The OED's earliest quotation is from 1560 and involves the phrase to "lay odds." Unlike the roughly contemporary word "probability," the word "odds" has always been associated with gambling. Almost as soon as there was a probability literature in English the word "odds" appeared in it. It appears only once in the translation (1714) of Huygens's tract on games of chance but is more common in De Moivre's The Doctrine of Chances (1718). However "odds" was not confined to works on gambling. It could be used wherever "probability" appeared. The two appear together in, e.g., Simpson's On the Advantage of Taking the Mean (1755) and Bayes's Essay (1763). These essays were not about gambling and nor was John Michell's "Inquiry into the Probable Parallax, and Magnitude of the Fixed Stars, from the Quantity of Light Which They Afford us, and the Particular Circumstances of Their Situation," Philosophical Transactions, 57, (1767), 234-264. This gave the odds that the Pleiades is a system of stars and not a random agglomeration. Probability authors have continued to use "odds" along with "probability" for variety and for dramatic effect.

This entry was contributed by John Aldrich. See PROBABILITY.

ODDS RATIO. The term was introduced by D. R. Cox in "The Regression Analysis of Binary Sequences," Journal of the Royal Statistical Society. Series B, 20, (1958), 215-242. The concept had been used by earlier authors, e.g. by R. A. Fisher in "The Logic of Inductive Inference," Journal of the Royal Statistical Society, 98, (1935), p. 50. [David (2001)].

OGIVE. The term was introduced in 1875 by Francis Galton in 'statistics by intercomparison with remarks on the Law of Frequency of Error.' Philos. Mag. 49, p. 35: "When the objects are marshalled in the order of their magnitude along a level base at equal distances apart, a line drawn freely through the tops of the ordinates..will form a curve of double curvature... Such a curve is called, in the phraseology of architects, an 'ogive'." The OED gives the architectural meaning as "A diagonal groin or rib of a vault, two of which cross each other at the vault's centre."

The term OMEGA RULE was first used by A. Grzegorczyk, A. Mostowski, A. and C. Ryll-Nardzewski in "The Classical and the w-Complete Arithmetic", JSL (1958), 188-206, according to Edgard G. K. López-Escobar e Ítala M. Loffredo D'Ottaviano, "A regra-w : passado, presente e futuro", Centro de lógica, epistemologia e história da ciéncia, Campinas-São Paulo, 1987.

An earlier term for this rule of inference, "Carnap's rule" was first used by J. Barkley Rosser in "Gödel-Theorems for Non-Constructive Logics (JSL 2 (1937), 129-37), where he alludes to Carnap's "Ein Gültigkeitskriterium für die Sätze der Klassischen Mathematik" (Monatsheffe für Mathematik und Physik 42 (1935), 163-90). However, as it was pointed in López-Escobar,
Infelizmente, isto está longe da realidade, porque Carnap não fala de uma regra de dedução que tenha qualquer semelhança com a regra-w .
[Unhappily, this is far from the reality, because Carnap doesn't speak about a deduction rule that has any resemblance with the w-rule.]
Another term for this rule, "Novikov's rule," was used by Schoenfield, Mostowski and Kreisel, who mention Novikov's "On the Consistency of Certain Logical Calculi" (Math. Sbornik 12 (1943), 231-61). But López-Escobar points out again (p. 3) that this name is inappropriate because Novikov's paper deals "only with infinite formulas with recursive conjunctions and disjunctions", not with the w-rule.

ONETH. This word is apparently not listed in any printed dictionary. However, it appears on numerous Web pages, in the phrase "n minus oneth" or "n plus oneth."

In 1890 Chambers's Encyclopaedia has, in the article on crystallography: "The plane 1 1 0 means a plane which cuts the x axis at one-oneth of a, the y axis at one-oneth of b, and the z axis at one-noughth of c—i. e. does not cut c at all, or is parallel to it." [Google print search]

ONE-TO-ONE CORRESPONDENCE is found in H. G. Zeuthen, "Sur les points fondamentaux de deux surfaces dont les points se correspondent un à un," C. R. LXX. 742. (1870).

One-to-one correspondence is found in English in 1873 in Proc. Lond. Math. Soc. IV. 252: "The equations .. being supposed to establish a 'one-to-one' correspondence between the two integral spaces." In his Principles of Mathematics Bertrand Russell (1903, p. 113) states, "Two classes have the same number...when their terms can be correlated one to one, so that any one term of either corresponds to one and one only term of the other." (OED).

See the entry INJECTION, SURJECTION and BIJECTION.

ONTO was used as a preposition in 1940 by C. C. MacDuffee in Introd. Abstract Algebra: "If a homomorphism of A onto B exists, we write A ~ B."

Onto was used as an adjective in 1942 by Solomon Lefschetz in Algebraic Topology: "If a transformation is 'onto,' ..." (page 7) (OED).

See the entry INJECTION, SURJECTION and BIJECTION.

OPEN SET. René Baire, "Sur les fonctions de variables réelles," Annali di Matematica Pura ed Applicata (3) 3 (1899), 1-123 has:
Je dirai que S est une sphère fermée, S' une sphère ouverte. La différence essentielle entre ces deux ensembles réside dans la fait suivant:

Etant donné un point quelconque de S', il existe une sphère ayant ce point pour centre et de rayon positif, dont tous les points font partie de S'. J'appelle d'une manière générale domaine ouvert à n dimensions tout ensemble de points possédant cette propriété.
A translation is:
I will say that S is a closed sphere and S' is an open sphere. The essential difference between these two sets resides in the following fact:

Given any point in S', there exists a sphere having this point for its center, and of positive radius, of which all its points are a part of S'. More generally, I will call an open domain in n dimensions any set of points that possesses this property.
This citation and translation were provided by Dave L. Renfro, who writes, "I believe that, in addition to being the first appearance of the term 'open set,' this may also be the first appearance of the concept itself. Prior to this, I think everyone just talked about the interior of a set of points, or about the 'sum' (i.e. union) of the interiors of sets from some specified collection of sets (often intervals)."

The phrase "open set of points" appears in W. H. Young, "On the density of linear sets of points," Proc. London Math. Soc. 34 (1902) [OED].

The concept of open set emerged with Lebesgue's 1905 paper on analytically representable functions. He wrote (p. 157): "les ensembles ouverts, c'est-à-dire ceux qui sont les complémentaires des ensembles fermés."

In a 29-30 December 1905 conference abstract [Bull. Amer. Math. Soc. 12 (1905-06), p. 285], Nels Johann Lennes wrote: "The expression 'entirely open region' is used to denote a set of points, no point of which is a limit point of points not in the set." This citation was provided by Dave L. Renfro, who writes that it may be the first appearance in English of the term with the meaning that it has today:

Open set is found in 1906 in W. H. Young and G. C. Young, Theory of Sets of Points: "The set of all the points inside a triangle is called a triangular domain, or the interior of the triangle, and is a simple case of a region .... The points of a domain always form an open set" [OED]. Here the term is used to mean "not closed" rather than "complement of closed" as Lebesgue, and mathematicians now, use the term.

Weierstrass used connected open sets in n-space and called such a set a "Gebiet."

Hausdorff in 1914 took over Weierstrass's term "Gebiet," but changed it to mean an open set, and explicitly referred to Weierstrass's earlier usage of the term to mean open connected set.

Open set is found in 1939 in M. H. A. Newman, Elem. Topology of Plane Sets of Points: "The sum of any set of open sets is an open set" [OED].

[Much of this entry was taken from a mailing list message by Gregory H. Moore.]

OPERAND appears in Sir William Rowan Hamilton, Lectures on Quaternions (London: Whittaker & Co, 1853): "...in order to justify the subsequent abstraction of the operand step-set, or the abridgement (compare ) of this formula of successive operation to the following..." [James A. Landau].

OPERATING CHARACTERISTIC CURVE appears in A. Wald "Sequential Tests of Statistical Hypotheses," Annals of Mathematical Statistics, 16, (1945), 117-186.

OPERATION. Christopher Clavius (1537-1612) used the term operationes in his Algebra Christophori Clavii Bambergensis of 1608 (Smith vol. 2).

In English operation appears in 1713 in Introd. Math. by J. Ward: "If the whole Æquation..be now taken, and we proceed to a Second Operation, the Value of a may be increas'd with twelve Places of Figures more, and those may be obtain'd by plain Division only [OED].

OPERATIONS RESEARCH (OPERATIONAL RESEARCH). The OED defines operational research as "a method of mathematically based analysis for providing a quantitative basis for management decisions (originally for military planning); abbreviated OR." Operational research is the British term, in the United States operations research is used. OR grew out of the work done by mathematical scientists in military planning in the Second World War.

The OED's earliest quotation is from a 1941 document written by P. M. S. Blackett, "The work of an Operational Research Section should be carried out at Command, Groups, Stations or Squadrons as circumstances dictate" published in Advancement of Science (1948), 5, 27/1 The first JSTOR sighting is an item of personal news in the Annals of Mathematical Statistics, 14, (Dec., 1943), p. 442: "Mr Blair M. Bennett is attached to the Operations Research Section of the Eighth Bomber Command."

In 1950 the British Operational Research Club created a quarterly journal. It is now called the Journal of the Operational Research Society. The Operations Research Society of America was formed in 1952 and its journal appeared that year. In 1956 the journal was renamed Operations Research.

ORBIT (in group theory). In a message posted to sci.math in 2009, Robert Israel wrote that Using MathSciNet the earliest reference he found to orbit that did not appear to be in the context of mechanics was Dick Wick Hall, "An example in the theory of pointwise periodic homeomorphisms," Bull. Amer. Math. Soc. 45 (1939), Part 1:882-885: "For each point x of M the finite subset of M consisting of all the images of x under T is called the orbit of x under T."

ORDER (degree) is found in English in 1706 in Ditton, Fluxions 22: "An Infinitesimal of another Order or Degree" and ibid. 123: "These sorts of [Exponential] Quantities are of several Orders or Degrees" [OED].

Gradus appears in the title De Constructione Aequationum Differentialium Primi Gradus (1707) by Gabriele Manfredi (1681-1761).

Degree is found in Dictionarium Britannicum (1730-6): "Parodic Degree (in Algebra) is the index or exponent of any power; so in numbers, 1. is the parodick degree, or exponent of the root or side; 2. of the square, 3. of the cube, etc" [OED].

Euler used gradus in the title "Nova methodus innumerabiles aequationes differentiales secundi gradus reducendi ad aequationes differentiales primi gradus," which was published in Commentarii academiae scientarum imperialis Petropolitanae, Tomus III ad annum 1728, Petropoli, 1732, pp. 124-137. The title is translated "A New Method of reducing innumerable differential equations of the second degree to differential equations of the first degree," although the modern terminology is "order" of a differential equation.

Order is found in English in 1743 in William Emerson, The Doctrine of Fluxions: "In any Fluxionary Equation, a Quantity of the first Order is that which has only one first Fluxion in it; a Quantity of the second Order has either one second Fluxion or two First Fluxions: Quantities of the third Order, are third Fluxions, product of three first Fluxions, product of a first and second Fluxion, etc." [OED].

ORDER OF MAGNITUDE appears in 1853 in A dictionary of science, literature & art: "It is therefore called the modulus of elasticity, or coefficient of elasticity; and is of the same nature and order of magnitude as the cohesive resistance which bodies oppose to rupture on being crushed, and may therefore be expressed as so many pounds acting on a square inch of surface" [University of Michigan Digital Library].

Order of magnitude appears in 1851-54 in Hand-books of natural philosophy and astronomy by Dionysius Lardner: "As spaces or times which are extremely great or extremely small are more difficult to conceive than those which are of the order of magnitude that most commonly falls under the observation of the senses" [University of Michigan Digital Library].

Order of magnitude appears in March 1857 in I. T. Danson, "Connection between American Slavery and the British Cotton Manufacture," Debow's review, Agricultural, commercial, industrial progress and resources: "The two customers next on the list, when arranged in order of magnitude, are the United States and the United Kingdom."

Order of magnitude appears in 1858 in The Plurality of Worlds by William Whewell: "Applying these considerations to the stars of inferior orders of magnitude, he finally arrives at the following conclusion, which he admits to be of an unexpected character:--that the number of insulated stars is indeed greater than the number of compound systems; but only three times, perhaps only twice as great."

For the big O and little o symbols see Earliest Uses of Symbols of Number Theory

ORDER OF OPERATIONS. In Feb. 1842 in "Remarks on the distinction between algebraical and functional equations" in the Cambridge Mathematical Journal, Robert Leslie Ellis wrote: "As the distinction between functional and common equations depends on the order of operations, it follows that, when part of the solution of an equation does not vary with the nature of the operation subjected to the resolving process, this part is applicable as much to functional equations as to any other" [University of Michigan Historical Math Collection].

Order of Arithmetical Operations is found as a section heading in the article Algebra in the 1911 Encyclopaedia Britannica.

Order of operations is found as a chapter heading in 1913 in Second Course in Algebra by Webster Wells and Walter W. Hart.

ORDERED PAIR occurs in "A System of Axioms for Geometry," Oswald Veblen, Transactions of the American Mathematical Society, 5 (Jul., 1904): "Each ordered pair of elements determines a unique element that precedes it, a unique element that follows it and a unique middle element."

ORDINAL. The earliest citation for this term in the OED is in 1599 in Percyvall's Dictionarie in Spanish and English enlarged by J. Minsheu, in which the phrase ordinall numerals is found.

ORDINARY DIFFERENTIAL EQUATION is found in 1828 in An Elementary Treatise on the Differential and Integral Calculus by Jean-Louis Boucharlat and Ralph Blakelock: "If x, instead of being a function of two variables x and y, should contain only x, this would be no more than an ordinary differential equation, which, being integrated, would give..." [Google print search]

ORDINATE. Cajori (1906, page 185) writes: "The Latin term for 'ordinate,' used by Descartes comes from the expression lineae ordinatae, employed by Roman surveyors for parallel lines.

Cajori (1919, page 175) writes: "In the strictly technical sense of analytics as one of the coördinates of a point, the word "ordinate" was used by Leibniz in 1694, but in a less restricted sense such expressions as "ordinatim applicatae" occur much earlier in F. Commandino and others."

Leibniz used the phrase "per differentias ordinatarum" in a letter to Newton on June 21, 1677 (Scott, page 155).

Leibniz used the term ordinata in 1692 in Acta Eruditorum 11 (Struik, page 272).

For the word in English the OED has a passage from 1706: H. Ditton An Institution of Fluxions p. 31 "'Tis required to find the relation of the Fluxion of the Ordinate to the Fluxion of the Abscisse."

See the entry ABSCISSA.

ORIGIN. Boyer (page 404) seems to attribute the term origin to Philippe de Lahire (1640-1718).

The term presumably appears in Sections Coniques by Marquis de l'Hospital, since the OED shows a use of the term in English in a 1723 translation of this work: "The Parabola infinitely extends itself more and more on each Side the Axis AP, beginning from the Origin."

ORTHOCENTER. An earlier term was Archimedean point.

According to John Satterly, "Relations between the portions of the altitudes of a plane triangle," The Mathematical Gazette 46 (Feb. 1962), pp. 50-51, Mathematical Note 2997:
0s a matter of historical interest our readers may be reminded that the term "Orthocentre" was invented by two mathematicians, Besant and Ferrers, in 1865, while out for a walk along the Trumpington Road, a road leading out of Cambridge toward London. In those days it was a tree-lined quiet road with a sidewalk, a favourite place for a conversational walk.
This information was provided by Emili Bifet, who reports that Satterly does not provide further reference.

Orthocentre is found in 1866 in Trilinear Co-ordinates by Norman Macleod Ferrers: "It may be convenient to call the point of intersection of the perpendiculars let fall from the angular points of a triangle on the opposite sides, the orthocentre of the triangle." [OED]

ORTHOGONAL is found in English in 1571 in A geometrical practise named Pantometria by Thomas Digges (1546?-1595): "Of straight lined angles there are three kindes, the Orthogonall, the Obtuse and the Acute Angle." (In Billingsley's 1570 translation of Euclid, an orthogon (spelled in Latin orthogonium or orthogonion) is a right triangle.) [OED].

The term ORTHOGONAL MATRIX was used in 1854 by Charles Hermite (1822-1901) in the Cambridge and Dublin Mathematical Journal, although it was not until 1878 that the formal definition of an orthogonal matrix was published by Frobenius (Kline, page 809).

ORTHOGONAL TRANSFORMATION was used in 1859 by George Salmon in Lessons Introductory to the Modern Higher Algebra: "What we may call the orthogonal transformation is to transform simultaneously a given quadratic function..." [OED].

ORTHOGONAL FUNCTION. Maxime Bôcher wrote in An Introduction to the Study of Integral Equations (p. 53): "So far as the writer knows, the term orthogonal was first used in this sense by F. Klein in a course of lectures on the differential equations of mathematical physics, delivered in Göttingen in the summer of 1889." [Information from Jan Peter Schäfermeyer]

ORTHOGONAL VECTORS. The term perpendicular was used in the Gibbsian version of vector analysis. Thus E. B. Wilson, Vector Analysis (1901, p. 56) writes "the condition for the perpendicularity of two vectors neither of which vanishes is A·B = 0." When the analogy with functions was recognised the term "orthogonal" was adopted. It appears, e.g.., in Courant and Hilbert's Methoden der Mathematischen Physik (1924).

See the entry VECTOR.

ORTHOGONAL. In Probability and Statistics the term "orthogonal" has acquired additional meanings. The Oxford Dictionary of Statistical Terms (Y. Dodge (ed.) (2003)) lists 5 primary senses and has entries for 10 distinct terms; (some of these are taken from mainstream mathematics.) The 6 entries of the more specialised Encyclopaedia of Design Theory come with the warning that "orthogonal" has "many different meanings."

In the early 20th century "orthogonal" was most often met in the phrase ORTHOGONAL FUNCTION or orthogonal polynomial.. In a 1922 paper, "The goodness of fit of regression formulae, and the distribution of regression coefficients" (J. Royal Statist. Soc., 85, 597-612) R. A. Fisher uses the term "orthogonal" in this way but he also uses it to mean stochastically independent, as when he writes "a and b are orthogonal functions, in that given the series of values of x observed, their sampling variation is independent." (p. 607). More recently the term has been used as a synonym for uncorrelated.

ORTHOGONAL DESIGN. F. Yates writes in "The Principles of Orthogonality and Confounding in Replicated Experiments," Journal of Agricultural Science, 23, (1933), 108-145. "Orthogonality is that property of the design which ensures that the different classes of effects to which the experimental material is subject shall be capable of direct and separate estimation without any entanglement." (David (2001)).

ORTHOGONAL ARRAY. According to (Y. Dodge (ed.) (2003)) the orthogonal array was introduced by C. R. Rao "Confounded Factorial Designs in Quasi-Latin Squares," Sankhyā, 7, (1946), 295-304.

OSCILLATING SERIES. In the third edition of A Treatise on Algebra (1892), Charles Smith writes, "Such a series is sometimes called an indeterminate or a neutral series." A footnote reads, "These series are however called divergent series by Cauchy, Bertrand, Laurent and others."

Oscillating series appears in G. H. Hardy, "On certain oscillating series," Quart. J. 38.

Oscillating series appears in 1893 in A Treatise on the Theory of Functions by James Harkness and Frank Morley: "Strictly speaking, an oscillating series is distinct from a divergent series, but it is usual to speak of non-convergent series as divergent."

Oscillating series appears in 1898 in Introduction to the theory of analytic functions by Harkness and Morley:
It is then permissible to introduce parentheses in an infinite series. It is not always permissible to remove them; for example in the series (1 - 1) + (1 - 1) + (1 - 1) + ....., each term is 0 and the limit is 0, but the series 1 - 1 + 1 - 1 + ... oscillates; if we agree to take always an even number of terms the limit is 0; if an odd number, the limit is 1.
A footnote says, "Most English text-books regard oscillating series as not divergent."

The term OSCULATING was used by Leibniz in 1686, Acta Eruditorum, 1686, 289-92 = Math. Schriften, 7, 326-29 (Kline, page 556).

The term OSCULATING PLANE was introduced by John Bernoulli (Kline, page 559).

The term OSCULUM is due to Huygens.

Osculum is found in English in 1728 as an entry in Chambers Cyclopaedia: "A Circle described on the Point C, as a Centre,..with the Radius of the Evolute M C, is said to osculate, kiss, the Curve described by Evolution in M; which point M is call'd by the Inventor Huygens, the Osculum of the Curve." [OED]

OUTER PRODUCT. See INNER PRODUCT.

OUTLIER. The word has existed since the 17th century but it seems to have been first used as a statistical term in the early 20th century, while its widespread use dates only from the 1960s. A JSTOR search found Karl Pearson writing in "On the Laws of Inheritance in Man," Biometrika, 2, (1903), p. 367 "The improbability of the normal distribution is, however, in all these cases chiefly due to a small lump of "outliers" at the "giant" end of the distribution." But even in 1903 the treatment of "doubtful" or "abnormal" observations was an established topic in the theory of least squares. See, e.g., p. 558 of Chauvenet's A Manual of Spherical and Practical Astronomy ... with an Appendix on the Method of Least Squares (4th edition, 1871).

A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - W - X - Y - Z