CONTINUITY (Lot. continvitas, from con tinuos, uninterrupted, from continere, to hold together, from corn-, together + icacre, to hold). ln geometry, a vital principle which asserts that if from the nature of a particular problem we would expect a certain number of solutions, then there will be the sonic number of solutions in every case, although some may be imaginary. E.g. a straight line and a circle in the same plane intersect in two points real, coincident or imaginary. The sum of the angles of a quadri lateral is a perigon whether the quad i lateral is convex, cross, or concave. In this ease, however. angles which have decreased and have passed through zero must be regarded as negative. By the principle of emitinuity theorems concerning real points or lines may be extended to imaginary points or lines. This change can take place only when some element of the figure passes through either a zero value or an infinite value; e g. rotate an asymptote of the hyperbola about the origin; before rotation it. cuts the curve in two infinite points; after rotation it cuts it in two real points or two imaginary points. In case of the real points rotate it still further, and these pass to infinity, and imaginary points oc cur. Many propositions of elementary geometry may be inferred from this principle. It was first stated by Kepler, emphasized by Boseovich, and put into acceptable form by Poneelet in his Traite des propric'tes projectives des figures (2d ed., Paris. 1865-66).
More generally continuity is a philosophical concept exemplified in space and time. It has been defined as a series of adjacent parts with common limits; as, infinite divisibility; that is, that however small the segment between points, a further division is possible; but in modern analysis continuity is the essential prop erty of a continuum. By a continuum is mider
stood a system or manifoldness of parts possessed in varying degree of a property A, such that between any two parts distant a finite length from each other an infinite number of other parts may be interpolated, of which those that are immediately adjacent exhibit only indefi nitely small differences with respect to the prop erty A. This is expressed by Cantor as a 'perfekt zusammenhiingeude Menge,' a perfect concatenation of points; e.g. all numbers ra tional and irrational in any interval form a continuum. A concatenation not perfect is called a scmi-continuum; e.g. the rational or the irrational numbers in any interval. A straight line is said to possess continuity.
By the continuity of the roots of an equation is meant that as a result of certain variations of the function, different pairs of roots may during the process become equal or imaginary, the total number always continuing the same— lot example given by Leibnitz. By the continuity of a function of ,r is meant the fact that in definitely small and continuous changes in the value of x between certain limits produce in definitely small and continuous changes in the function. Consult : Jordan, ('ours d'amtlyse (Paris. 1893) Poncelet. Traits des proprieWs projeetires des figures (Paris, 1865-66) ; Epley elopudie der mathematischen Trissenschaften, vol. i. (Leipzig. 1901) ; Cantor, Mathematisehe Annalen, vols. xx. and xxi. (Leipzig, 1882-83) ; Mach, in The Open Court, vol. xiv. (Chicago, 1900).