RATIFICATION

(Let retain, determined, facio, / make.) The solemn act by which a competent authority Idlty to an instrument, agreement, . The term U ordinarily used In International law for the sanction governments to treaties contracted bv their representatives. In Preach law, ratification is defined the a;>- probation or confirmation of what has been done or promised. Thus, in many instances, a person, on attaining his majority, ratifies acts done by himself Of his guardian in his minority. And ratification is either expre ,,r tacit; the latter resulting, by implication, from his silence for ten:ter attaining his majority. B \ TIO (Lnt.), in Geometry, is defined bv Euclid (Elem ent.t, book v., def. 3) to be "a mutual relation of two magnitudes of the same kind to one another in n quantity." This definition has been much erltjd Barrow (Lectienee Math.) calls it a metaphysical definition, inasmuch as it is not properly a mathematical definition, since nothing in mathematics depends on it, or can be deduced from it; and supposes that Euclid had probably no other design in making it than to give a general summary idea of rario to beginners. It has been remarked, that as the word quantity in our language, if not quite synonymous with magnitude, has a signification only a little more general, the definition as above rendered is either tautological or unmeaning. Or. Bimsoa supposes it to be the interpolation of some unskilful editor. Leslie (Elements of Geome- •niously supposes that the Greek word TrnXtKOTva, which is usually translated quantity, may have reference tude or number, as well as to magnitude, and that Euclid's definition maybe rendered as follows: "Ratio is a certain mutual habitude of two homogeneous magnitudes with respect to quotity, or numerical composition." Dr. Will is (Opera Mathematic a, torn, ii., p. 6(15) translates the himc word by the Latin qua.ntuplir.itas, which refers to the number of times the one magnitude is contained in the -ither. Dr. Peacock (..1/grbra, p. 309) remarks, that there is no geometrical definition of ratio by which the equivalence of different modes of representation may be ascertained as nece ary consequences; and for this reason, ratios in geometry are only considered in connexion with enrh other, as constituting, or not constituting, a proportion. in arithmetic and algebra, a ratio may be defined as the [s. 1045]