Now the superabundant number is one which has, over and above the factors which belong to it and fall to its share, others in addition, just as if an animal should be created with too many parts or limbs, with 10 tongues as the poet says, and 10 mouths, or with 9 lips, or 3 rows of teeth, or 100 hands, or too many fingers on one hand. Similarly, if when all the factors in a number are examined and added together in one sum, it proves upon investigation that the number's own factors exceed the number itself, this is called a superabundant number, for it oversteps the symmetry which exists between the perfect and its own parts. Such are 12, 24, and certain others, . . .

Imperfecte nombers be suche, whose partes added together, doe make either more or lesse, then the whole nomber it self... And if the partes make more then the whole nomber, then is that nomber called superfluouse, or abundaunt (OED2).

Here tells þat þer ben .7. spices or partes of (th)is craft. The first is called addicion, þe secunde is called subtraccion. The thyrd is called duplacion. The 4. is called dimydicion. The 5. is called multiplicacion. The 6. is called diuision. The 7. is called extraccion of þe rote.

[Fermat] introduces the technical term adaequalitas, adaequare, etc., which he says he has borrowed from Diophantus. As Diophantus V.11 shows, it means an approximate equality, and this is indeed how Fermat explains the word in one of his later writings.

When Chapter I was written..I had not seen Forsyth's memoir, and had not been able to find an adopted English term for Lagrange's équation adjointé, so I used the word adjunct, suggested by the German adjungirte, and not unlike the French adjointe. It seems better now, however, to employ the word associate, or, when speaking simply of Lagrange's équation adjointé, the word adjoint.

The road leading from the algebra of logic to algebraic logic is an interesting object of study for the historian of modern logic which has yet to be fully examined. Curry stands in the middle of the road, he was the first to use the expression "algebraic logic" in (Curry 1952) and not Halmos as erroneously stated in (Blok/Pigozzi 1991, p. 365), but what he meant by it was still close to algebra of logic. Halmos introduced this expression rather to denote the algebraic treatement of first-order logic, but nowadays the expression "algebraic logic" is used to include both the zero and the first-order levels.

Ex cognito hoc velut Algorithmo, ut ita dicam, calculi hujus, quem voco differentialem, omnes aliae aequationes differentiales inveniri possunt per calculum communem, maximae que & minimae, item que tangentes haberi, ita ut opus non sit tolli fractas aut irrationales, aut alia vincula, quod tamen faciendum fuit secundum Methodos hactenus editas. (From this rule, known as an algorithm, so to speak, of this calculus, which I call differential, all other differential equations may be found by means of a general calculus, and maxima and minima, as well as tangents [may be] obtained, so that there may be no need of removing fractions, nor irrationals, nor other aggregates, which nevertheless formerly had to be done in accordance with the methods published up to the present.)

AGO, GOC, we have already named interior angles on the same side; BGO, GOD, have the same name; AGO, GOD, are called alternate interior angles, or simply alternate; so also, are BGO, GOC: and lastly EGB, GOD, or EGA, GOC, are called, respectively, the opposite exterior and interior angles; and EGB, COF, or AGE, DOF, the alternate exterior angles.

The English word amenable was intended by M. Day, who introduced the term, to remind the audience of the word "mean" since the concept of amenability is defined by the existence of a certain mean. Now, "mean" (in the mathematical sense) is "Mittel" in German, and "moyenne" in French. Accordingly, amenable (semi-)groups were called "mittelbar" in German and "moyennable" in French, since about 1950. J.-P. Pier, author of the book Amenable Locally Compact Groups, called them "groupes moyennables" in his earlier publications in French. The German word "mittelbar" might be called a barbarism, just like your characterization of the non-existing word "meanable" which may have induced Day to coin the term "amenable". However, "mittelbar" is certainly more suitable than von Neumann's "messbar" which usually means measurable.

The mathematicians, I grant you, have done their best to promulgate the popular error to which you allude, and which is none the less an error for its promulgation as truth. With an art worthy a better cause, for example, they have insinuated the term 'analysis' into application to algebra. The French are the originators of this particular deception; but if a term is of any importance -- if words derive any value from applicability -- then 'analysis' conveys 'algebra' about as much as, in Latin, 'ambitus' implies 'ambition', 'religio' 'religion', or 'homines honesti', a set of honorable men.

According to Boyer, the terminology "analytic geometry" was used probably for the first time by Michel Rolle in his "De l'evanouissement des quantitez inconnues dans la geometrie analytique," Mémoires de l'Académie royale des sciences (Paris), Année 1709 (1709. 9. Aoust), pp. 419-450. See Carl B. Boyer, History of Analytic Geometry (New York, 1956), p. 155: "Incidentally, in Rolle's discussion of this problem, the term 'analytic geometry' appeared in print, perhaps for the first time, in a sense analogous that of today." On p. 141 of the same book Boyer also refers to Newton's treatise titled "Artis Analyticae Specimina vel Geometria Analytica," which was published in Operae quae exstant omnia, Vol. I, pp. 389-518, under the editorship of Samuel Horsley in London in 1779. The title is ascribed not to the author Newton but to a copyist William Jones. The tract which Horsley edited as Artis analyticae specimina vel Geometria analytica is basically identical to "De Methodus serierum et fluxionum," written in Winter 1670-71, in D. T. Whiteside, ed, The Mathematical Papers of Isaac Newton, Vol. III (1670-1673), (Cambridge, 1969). Jones copied Newton's manuscript about 1710. See Whiteside's Introduction, Op. cit., pp. 11-13. See also Whiteside's "General Introduction," in The Mathematical Papers of Isaac Newton, Vol. I (1664-1666), (Cambridge, 1967), pp. xxiii and xxviii, notes (26) and (34). I am skeptical about Newton's use of "analytic geometry" in other works as well.

It is absolutely intolerable to use analytical geometry for linear algebra with coordinates, still called analytical geometry in the elementary books. Analytical geometry in this sense never existed. There are only people who do linear algebra badly, by taking coordinates and this they call analytical geometry. Out with them! Everyone knows that analytical geometry is the theory of analytical spaces, one of the deepest and most difficult theories of all mathematics.

Prop. IX. Problem: A Projection being made as you please, having the Distance and Altitude, or Descent of an Object, through which the Project passes, together with the Angle of Elevation of the line of Direction; to find the Parameter and Velocity, that is (in Fig. 2.) having the Angle FGB, GM, and MX.

And they are also the Logarithmes of the complements of the arches and sines towards the right hand, which we call Antilogarithmes.

The buildings considered in this talk are particular simplicial complexes naturally associated to algebraic simple groups. The "real estate" terminology due to N. Bourbaki [éléments de mathématique. Groupes et algèbres de Lie ch. 4-6. 1968], originated in the fact that the maximal simplexes of these complexes are called "chambers" (in French, "chambre", that is, "room") because of their close connection with the "Weyl chamber" in the theory of root systems.

... we have a collection of Lemmas (Liber Assumptorum) which has reached us through the Arabic. [...] The Lemmas cannot, however, have been written by Archimedes in their present form, because his name is quoted in them more than once.

... though it is quite likely that some of the propositions were of Archimedean origin, e.g. those concerning the geometrical figures called respectively [arbelos, in Greek] (literally 'shoemaker's knife') and [salinon, in Greek] (probably a 'salt-cellar'), ...

We read it, "x equals the arc whose sine is a, or more briefly, "arc-sine a," i. e. x equals arc-sine a. The expression is sometimes read "x equals the inverse sine of a," or "the anti-sine of a."

The word "arithmetic," like most other words, has undergone many vicissitudes. In the Middle Ages, through a mistaken idea of its etymology, it took an extra r, as if it had to do with "metric." So we find Plato of Tivoli, in his translation (1116) of Abraham Savasorda, speaking of "Boetius in arismetricis." The title of the work of Johannes Hispalensis, a few years later (c. 1140), is given as "Arismetrica," and fifty years later than this we find Fibonacci dropping the initial and using the form "Rismetirca." The extra r is generally found in the Italian literature until the time of printing. From Italy it passed over to Germany, where it is not uncommonly found in the books of the 16th century, and to France, where it is found less frequently. The ordinary variations in spelling have less significance, merely illustrating, as in the case with many other mathematical terms, the vagaries of pronunciation in the uncritical periods of the world's literatures.

Take an Arithmetical mean between GH and CD, Diameters at the Top and Bottom of the Crown, viz. VW, and multiply the Area answerable thereto by IF the Altitude of that Segment, the Product is the Content of GHDC from which subduct the Content of the Crown CFD, the remainder is the Liquor lying about the Crown when just covered, which added to the upper part, the Sum is the Content of the whole Copper.

In diesem dritten Artikel (die frueheren s.F.d.M. X. 34, JFM10. 0034.02) setzt Herr McColl seine zuerst unter dem Namen "Symbolical Language" veroeffentlichte und zunaechst zur Verwendung auf mathematische Probleme bestimmte Arithmetisirung der Logik fort und bringt sie zu einem gewissen Abschluss. (JFM11.0049.01)

... ich glaube auch, dass es dereinst gelingen wird, den gesammten Inhalt aller dieser mathematischen Disciplinen zu "arithmetisiren", d.h. einzig und allein auf den im engsten Sinne genommenen Zahlbegriff zu gruenden, also die Modificationen und Erweiterungen dieses Begriffs wieder abzustreifen ...

However, in virtue of the same definitions, it will be found that another important property of the old multiplication is preserved, or extended to the new, namely, that which may be called the associative character of the operation....

This paper is marked as having been communicated on Nov. 13, 1843, but it closed with a note that it had been abstracted from a larger paper to appear in the Transactions of the academy (Math. Papers, 3:116). That longer paper, also dated Nov. 13, 1843 was not published until 1848, and when it did appear it concluded with a note mentioning works written as late as June 1847 ("Researches respecting Quaternions: First Series," Royal Irish Academy, Transactions 21, [1848]: 199-296, in Math. Papers, 3:159-216, "note A," pp. 217-26). In a note Hamilton stated that his presentation on Nov. 13, 1843 had been "in great part oral" (Math. Papers, 3:225); therefore it is possible that his statement of and naming of the associative law was added later when his original communication was to be printed in 1844. It is thus possible that he first recognized the importance of the associative law in 1844, when he began work on Graves's octaves.

The astroid seems to have acquired its present name only in 1838, in a book published in Vienna; it went, even after that time, under various other names, such as cubocycloid, paracycle, four-cusp-curve, and so on. The equation $x^{2/3} + y^{2/3} = a^{2/3}$ can, however, be found in Leibniz's correspondence as early as 1715.

Bekanntlich sind $\lambda$ und $√f(\lambda)$ in dem soeben gefundenen $\eta$ eindeutig; sie stellen solche eindeutige Functionen von $\eta$ vor, welche sich bei unendlich vielen linearen Substitutionen von η reproduciren (ich möchte vorschlagen, solche Functionen überhaupt automorphe Functionen von $\eta$ zu nennen).

Bei diesen bleiben dann, $\lambda_{1}, \lambda_{2}$ ungeändert; ich schlage dementsprechend vor, dieselben als automorphe Formen von $F_{1}, F_{2}$ zu bezeichnen.

Hr. Poincaré hat über die gedachten Functionen ausser einer grossen Reihe kleinerer mit dem Jahre 1881 beginnender Noten in der Comptes Rendus zunächst 1881 einen Aufsatz in den Mathematischen Annalen (Bd. 19: Sur les fonctions uniformes qui se reproduisent par des substitutions linéaires) .. Hr. Klein bezeichnet die hier in Betracht kommenden Functionen neuerdings als linear-automorphe Functionen oder wohl auch kurzweg als automorphe Functionen.

The word "avoirdupois" is more properly spelled "averdepois," and it so appears in some of the early books. It comes from the Middle English aver de poiz, meaning "goods of weight." In the 16th century it was commonly called "Haberdepoise," as in most of the editions of Recorde's (c. 1542) Ground of Artes. Thus in the Mellis eddition of 1594 we have: "At London & so all England thorugh are vsed two kids of waights and measures, as the Troy waight & the Haberdepoise."

The present proof rests upon the assumption that coverings gamma actually do exist, hence upon the principle that even for an infinite totality of sets there are always mappings that associate with every set one of its elements, or, expressed formally, that the product of an infinite totality of sets, each containing at least one element, itself differs from zero. This logical principle cannot, to be sure, be reduced to a still simpler one, but it is applied without hesitation everywhere in mathematical deduction.

Imagine that with every subset M' there is associated an arbitrary element m'₁ that occurs in M' itself...This yields a "covering" gamma of the [set of all subsets M'] by certain elements of the set M.

AXIOM VI. (Axiom of choice). If T is a set whose elements all are sets that are different from 0 and mutually disjoint, its union "union of T" includes at least one subset S₁ having one and only one element in common with each element of T. [The original German read "Axiom der Auswahl".]

Now in order to apply our theorem to arbitrary sets, we require only the additional assumption that a simultaneous choice of distinguished elements is in principle always possible for an arbitrary set of sets ...

IV. Axiom. A set S that can be decomposed into a set of disjoint parts A, B, C..., each containing at least one element, possesses at least one subset S₁ having exactly one element in common with each of the parts A,B,C,... considered.

Even Peano's Formulaire, which is an attempt to reduce all of mathematics to "syllogisms" (in the Aristotelian-Scholastics sense), rests upon quite a number of unprovable principles; one of these is equivalent to the principle of choice for a single set and can then be extended syllogistically to an arbitrary finite number of sets.