form apparent unities. Before a step could be made to higher problems, it was necessary to see that the proximate truth of the qualitative world,--or world of sense proper (ἰδία αἴσθησις) is in its simplest terms a quantitative world, or world of common sensibles (κοινὰ αἴσθητα), universalised sensibles, number and quantity. The sphere of quantity need only be briefly sketched. It has its three heads: (1) quantity in general,--the universal and vague notion of quantitativeness, the mere conception of reality as the Great and the Little, or the More and Less[12]: (2) Quantum, or defined quantity, expressed in the shape of a number: and (3) the quantitative ratio or degree, which is the individualisation or self-determination of numbers, or their application to one another,--which gives the real meaning and value of numbers. The fundamental antithesis, which we found in quality, comes before us here more definitely as the opposition of many ones in one number. In every quantity there are the two elements: the 'one,' unity or solidarity, which renders a total number possible, and the 'many' or multiplicity, which gives it real body and character. By this quantitative law, reality must always be both Continuous and Discrete. Thus when I regard a line as consisting of a number of points I treat it as a discrete quantity: as many in one. When, on the other hand, I regard the line as the unity of these points, it becomes a continuous quantity. These distinctions are not so trivial as they may appear: they lie at the bases of paradoxes like those by which Zeno disproved the ordinary representations of motion, and when a M.P. informs the House of Commons that it is impossible to divide 73l. 1s. 6d. by 1l. 2s. 6d., he is, like Zeno, and perhaps more unconsciously, forgetting that these quantities are not merely continuous but discrete. The Pythagoreans, according to the tradition of antiquity, philosophised number. In it they found the reality, or the principle of things,--the characteristic feature which dominated existence, and by which the world in all its multiplicity could be made coherent and intelligible. They saw it composed of two elements: a limit or limiting, and an unlimited: the latter as it were a dark ground, measureless and endless, on which definiteness was gradually marked out. Such a limiting principle would be e. g. the unit of number. But the full definiteness of number only comes out when a numerical scale is fixed on, in which each number bears a definite ratio to what goes before and what comes after. Each number in such a scale is really a multiple of its unit: a product of its unity into its multeity, of the monad into the indefinite duad. It is this view of each number, as the product of its prime unit with the ratio, which comes explicitly to the fore in Degree, or quantitative ratio. Each so-called quantitative statement is thus a ratio between a given quantity in the object and an assumed standard or unit of number.