 Following Weierstrass’s efforts, another attack on the problem of formulating rigorous definitions of continuity and the real numbers was mounted by Richard Dedekind (1831–1916). Dedekind focussed attention on the question: exactly what is it that distinguishes a continuous domain from a discontinuous one? He seems to have been the first to recognize that the property of density, possessed by the ordered set of rational numbers, is insufficient to guarantee continuity. In Continuity and Irrational Numbers (1872) he remarks that when the rational numbers are associated to points on a straight line, “there are infinitely many points [on the line] to which no rational number corresponds” (1872: section 3 [1999: 770]) so that the rational numbers manifest “a gappiness, incompleteness, discontinuity”, in contrast with the straight line’s “absence of gaps, completeness, continuity” (1872: section 3 [1999: 771]). Dedekind regards this principle as being essentially indemonstrable; he ascribes to it, rather, the status of an axiom “by which we attribute to the line its continuity, by which we think continuity into the line” (1872 [1999: 771–772]. It is not, Dedekind stresses, necessary for space to be continuous in this sense, for “many of its properties would remain the same even if it were discontinuous” (1872 [1999: 772]).