*Oblique angle* appears in English in 1695 in *Geometry
Epitomized* by William Alingham: "An Oblique Angle, is either
Acute or Obtuse" (OED2).

**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" (OED2).

*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** is dated 1639 in MWCD10.

**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" (OED2).

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" (OED2).

**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 Novik'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 dictionary. However,
it appears on numerous Web pages, in the phrase "*n* minus oneth" or "*n* plus oneth."

**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 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

quelconquede 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 généraledomaine ouvertà ndimensionstout 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

anypoint 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 anopen domainin ndimensionsany 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) (OED2).

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. & 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" (OED2). 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" (OED2).

[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 [25]) 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 (OED2).

**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.*”

*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" (OED2).

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." (OED2).

**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 OED2 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. Commandinus 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 OED2 shows a use of the term in English in a
1723 translation of this work.

**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:

This information was provided by Emili Bifet, who reports that Satterly does not provide further reference.Note:As 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.

*Orthocenter* is found in 1869 in *Conic sections, treated geometrically*
by William Henry Besant (1828-1917):
"If a rectangular hyperbola circumscribe a triangle, it passes through the orthocentre" (OED2).

The 1895 ninth edition of this work (and presumably the first edition also) has: "If a rectangular hyperbola circumscribe a triangle, it passes through the orthocentre. Note: The orthocentre is the point of intersection of the perpendiculars from the angular points on the opposite sides."

**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.) (OED2).

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..." (OED2).

**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.

**OUTER PRODUCT.** See INNER PRODUCT.

**OUTLIER.** The word has existed since the 17^{th} century
but it seems to have been first used as a statistical term in the early 20^{th}
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 (4^{th} edition, 1871).