[58]. [A more familiar example of this is the introduction of irrational or absurd numbers like √2. After it was proved that no ratio of two integers could possibly equal √2 the idea of number was generalized to include the latter. Fractions and the so-called imaginary numbers illustrate the same process of generalization for the sake of making certain operations (i.e. division and finding the root) continuously applicable.]
[59]. The Monist, April, 1892.
[60]. Continuous is not exactly the right word, but I let it go to avoid a long and irrelevant discussion.
[61]. The Monist, July, 1892.
[62]. This proposition is substantially the same as a theorem of Cantor, though it is enunciated in a much more general form.
[63]. The Monist, October, 1892.
[64]. I am rejoiced to find, since my last paper was printed, that a philosopher as subtle and profound as Dr. Edmund Montgomery has long been arguing for the same element in the universe. Other world-renowned thinkers, as M. Renouvier and M. Delbœuf, appear to share this opinion.
[65]. By a vera causa, in the logic of science, is meant a state of things known to exist in some cases and supposed to exist in other cases, because it would account for observed phenomena.
[66]. Wiedemann, Annalen, 1887-1889.
[67]. See Maxwell on Spherical Harmonics, in his Electricity and Magnetism.