Oblique angle appears in English in 1695 in Geometry Epitomized by William Alingham: "An Oblique Angle, is either Acute or Obtuse" (OED2).
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).
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."
OCTAGON is dated 1639 in MWCD10.
ODD FUNCTION is found in 1849 in Trigonometry and Double Algebra by Augustus De Morgan: "But the sine is what is called an odd function; that is, it changes sign when [theta] is changed into -[theta]; or sin (-[theta] = -sin [theta]" [University of Michigan Historical Math Collection].
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.
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.
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" (OED2).
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).
OPEN SET. 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 overts, c'est-a-dire ceux qui sont les complementaires des ensembles fermes."
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 Legesgue, and mathematicians now, use the term.
Weierstrass used connected open sets in n-space and called such a set a "Gebiet."
Open set is found in 1902 in Proc. Lond. Math. Soc. XXXIV. 289 in the heading, "Open sets of points" (OED2).
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].
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).
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" (OED2).
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."
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 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.
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).
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:
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.This information was provided by Emili Bifet, who reports that Satterly does not provide further reference.
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).
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.