When an expression raised to the square or any higher power vanishes, it may be called nilpotent; but when, raised to a square or higher power, it gives itself as the result, it may be called idempotent. The defining equation of nilpotent and idempotent expressions are respectively $A^{n} = 0$, and $A^{n} = A$; but with reference to idempotent expressions, it will always be assumed that they are of the form

$A^{2} = A$,

unless it be otherwise distinctly stated.

My most nearly immortal contributions are an abbreviation and a typographical symbol. I invented "iff", for "if and only if"—but I could never believe that I was really its first inventor. I am quite prepared to believe that it existed before me, but I don't know that it did, and my invention (re-invention?) of it is what spread it through the mathematical world. The symbol is definitely not my invention—it appeared in popular magazines (not mathematical ones) before I adopted it, but, once again, I seem to have introduced it into mathematics. It is the symbol that sometimes looks like▐, and is used to indicate an end, usually the end of a proof. It is most frequently called the "tombstone", but at least one generous author referred to it as the "halmos".

...tant les vrayes racines que les fausses ne sont pas tousiours réelles; mais quelquefois seulement imaginaires; c'est à dire qu'on peut bien toujiours en imaginer autant que aiy dit en chàsque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde à celles qu'on imagine. comme encore qu'on en puisse imaginer trois en celle cy, $x^{3} - 6xx + 13x - 10 = 0$, il n'y en a toutefois qu'une réelle, qui est 2, & pour les deux autres, quois qu'on les augmente, ou diminué, ou multiplié en la façon que ie viens d'éxpliquer, on ne sçauroit les rendre autres qu'imaginaires.
[...neither the true roots nor the false are always real; sometimes they are, however, imaginary; namely, whereas we can always imagine as many roots for each equation as I have predicted, there is still not always a quantity which corresponds to each root so imagined. Thus, while we may think of the equation $x^{3}- 6xx + 13x - 10 = 0$ as having three roots, yet there is just one real root, which is 2, and the other two, however, increased, diminished, or multiplied them as we just laid down, remain always imaginary.] (page 380)

Dic Angulum Contactûs esse, Infinitesimam partem duorum Rectorum: (seu 2/∞ R.)
[Say the angle of contact is an infinitesimal part [pars infinitesima] of two right angles (or 2/∞ R.)]

Linea, ex infinitis punctis, hoc est, Lineolis infinite exiguis, longitudine aequalibus, vel aeque altis; quarum cujusvis longitudo vel altitudo sit 1/∞ (pars infinitesima) longitudinis vel altitudinis totius lineae

Quia tamen sæpius curvatura mutatur, ut in conchoide Nicomedea, quæ pertinet ad priorem casum, et in omnibus speciebus curvæ Domini de Roberval (prima excepta) quæ pertinet ad secundum, ut perfecte curva possit delineari, investiganda sunt ex arte puncta inflexionum, in quibus curvatura ex convexa fit concava vel contra: cui negotio eleganter inservit doctrina de maximis et minimis, hoc prwmisso lemmate generali:

vous treuverez qu'il y a un certain point en icelle comme B, tel que depuis A iusques en B, elle est convexe en dehors, et depuis B par C à l'infiny, elle est convexe en dedans, ce qui est admirable. Et encor plus que par le point B, il ne se peut mener de ligne droite qui la touche, mais une comme OBY, qui fera les angles au sommet OBA, YBC, chacun moindre qu'aucun angle rectiligne donne. Il en sera de mesme de l'autre part, et ce sont ces deux points d'où ie vous mandois qu'on ne pouvoit mener de tangentes. (p. 45)

Another useful function of Bourbaki's treatise has been to standardise notation and terminology… Standard terms are badly needed for "one-to-one," "onto" and "one-to-one onto"; will Bourbaki's "injection," "surjection" and "bijection" prove acceptable?

When the German Sanskrit scholar Hermann Günther Grassman (1809-1877) developed the general algebra of hypercomplex numbers, he realized that more than one type of multiplication is possible. To two of the many possible types he gave the names inner and outer. The names seem to have been chosen because they are antonyms rather than for any intrinsic meaning.

Of nombres one is lyneal, ano(th)er superficialle, ano(th)er quadrat, ano(th)cubike or hoole.

At one time Leibniz and Johann Bernoulli discussed in their letters both the name and the principal symbol of the integral calculus. Leibniz favored the name calculus summatorius and the long letter [long S symbol] as the symbol. Bernoulli favored the name calculus integralis and the capital letter I as the sign of integration. ... Leibniz and Johann Bernoulli finally reached a happy compromise, adopting Bernoulli's name "integral calculus," and Leibniz's symbol of integration.

On what grounds the conviction of the unassailable exactness of mathematical laws is based has for centuries been an object of philosophical research, and two points of view may here be distinguished, intuitionism (largely French) and formalism (largely German). ... The question where mathematical exactness does exist, is answered differently by the two sides; the intuitionist says: in the human intellect, the formalist says: on paper. (pp. 82-3, OED)

To every point in space, it has been remarked, will correspond one particular family of equations all of the same character as regards the number they contain of real or imaginary roots, because capable of being derived from one another by real linear substitutions, such family consisting of an infinite number of ordinary or conjugate equations according as the point is facultative or non-facultative; but it may be well to notice that, conversely, every point does not correspond to a distinct family. In fact every point in the curves Johann Bernoulli (p, q being constants) will denote a curve divided into two branches by the origin of coordinates, one of which will be facultative and the other non-facultative; but in each separate branch every point will represent the very same family. Any such separate branch may be termed an isomorphic line; and we see that the whole of space may be conceived as permeated by and made up of such lines radiating out from the origin in all directions.

67. Un groupe Γ est dit isomorphe à un autre groupe G, si l'on peut établir entre leurs substitutions une correspondance telle : i° que chaque substitution de G corresponde à une seule substitution de Γ, et chaque substitution de Γ à une ou plusieurs substitutions de G; 2° que le produit de deux substitutions quelconques de G corresponde au produit de leurs correspondantes respectives. [Google print search]