Note 37 — Introductory material

With this coordinate system, then, to define distance as a certain function of the coordinates is not to be guilty of a vicious circle. But it by no means follows that the definition of distance is arbitrary. All reference to distance has been hitherto excluded, to avoid metrical ideas; but when distance is introduced, metrical ideas inevitably reappear, and we have to remember that our coordinates give no information, primâ facie, as to any of these metrical ideas. It is open to us, of course, if we choose, to continue to exclude distance in the ordinary sense, as the quantity of a finite straight line, and to define the word distance in any way we please. But the conception, for which the word has hitherto stood, will then require a new name, and the only result will be a confusion between the apparent meaning of our propositions, to those who retain the associations belonging to the old sense of the word, and the real meaning, resulting from the new sense in which the word is used.