早期數學字彙的歷史 (O)

Last revision: June 22, 1999


OBTUSE ANGLE appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements: "An acute angle is that, which is lesse then a right angle"; "an obtuse angle is that which is greater then a right angle" (OED2).

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. L鏕ez-Escobar e 疘ala M. Loffredo D'Ottaviano, "A regra-w : passado, presente e futuro", Centro de l鏬ica, epistemologia e hist鏎ia da ci幯cia, Campinas-S緌 Paulo, 1987.

An earlier term for this rule of inference, "Carnap's rule" was first used by J. Barkley Rosser in "G鐰el-Theorems for Non-Constructive Logics (JSL 2 (1937), 129-37), where he alludes to Carnap's "Ein Gltigkeitskriterium fr die S酹ze der Klassischen Mathematik" (Monatsheffe fr Mathematik und Physik 42 (1935), 163-90). However, as it was pointed in L鏕ez-Escobar,

Infelizmente, isto est longe da realidade, porque Carnap n緌 fala de uma regra de dedu誽o que tenha qualquer semelhan蓷 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 L鏕ez-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 co顤dinates 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 Gonz嫮ez Cabill鏮).

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.


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