**OCTAGON** is dated 1639 in MWCD10.

**ODD FUNCTION** is found in 1886 in *Differential and Integral
Calculus* by A. G. Greenhill (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).

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.

**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 S. Lefschetz
in *Algebraic Topology*: "If a transformation is 'onto,' ..." (OED2).

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

**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.

**ORDINATE.** 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 term *ordinata* in 1692 in *Acta Eruditorum
11* (Struik, page 272).

**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** was coined in 1869 by William Henry Besant (1828-1917)
in his *Conic sections, treated geometrically,* London: G. Bell and
Sons, 1869 (Julio González Cabillón).

**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).

The term **OSCULATING** was used by Leibniz in 1686, *Acta Eruditorum,*
1686, 289-92 = *Math. Schriften,* 7, 326-29 (Kline, page 556). John
Bernoulli introduced the term *osculating plane* (Kline, page 559).
The early term **OSCULUM** is due to Huygens.

**OUTER PRODUCT.** See *inner product.*