In every demonstration three things may be distinguished: (1) The demonstrated conclusion, or Attribute essential to a certain genus; (2) The Genus, of which the attributes _per se_ are the matter of demonstration; (3) The Axioms, out of which, or through which, the demonstration is obtained. These Axioms may be and are common to several genera: but the demonstration cannot be transferred from one genus to another; both the extremes as well as the middle term must belong to the same genus. An arithmetical demonstration cannot be transferred to magnitudes and their properties, except in so far as magnitudes are numbers, which is partially true of some among them. The demonstrations in arithmetic may indeed be transferred to harmonics, because harmonics is subordinate to arithmetic; and, for the like reason, demonstrations in geometry may be transferred to mechanics and optics. But we cannot introduce into geometry any property of lines, which does not belong to them _quâ_ lines; such, for example, as that a straight line is the most beautiful of all lines, or is the contrary of a circular line; for these predicates belong to it, not _quâ_ line, but _quâ_ member of a different or more extensive genus.[26] There can be no complete demonstration about perishable things, or about any individual line, except in regard to its attributes as member of the genus line. Where the conclusion is not eternally true, but true at one time and not true at another, this can only be because one of its premisses is not universal or essential. Where both premisses are universal and essential, the conclusion must be eternal or eternally true. As there is no demonstration, so also there can be no definition, of perishable attributes.[27] [Footnote 26: Ibid. vii. p. 75, a. 38-b. 20. Mr. Poste, in his translation, here cites (p. 50) a good illustrative passage from Dr. Whewell's Philosophy of the Inductive Sciences, Book II. ii.:--"But, in order that we may make any real advance in the discovery of truth, our ideas must not only be clear; they must also be _appropriate_. Each science has for its basis a different class of ideas; and the steps which constitute the progress of one science can never be made by employing the ideas of another kind of science. No genuine advance could ever be obtained in Mechanics by applying to the subject the ideas of space and time merely; no advance in Chemistry by the use of mere mechanical conceptions; no discovery in Physiology by referring facts to mere chemical and mechanical principles." &c.] [Footnote 27: Aristot. Analyt. Post. I. viii. p. 75, b. 21-36. Compare Metaphys. Z. p. 1040, a. 1: [Greek: dê=lon o(/ti ou)k a)\n ei)/ê au)tô=n (tô=n phthartô=n) ou)/th' o(rismo\s ou)/t' a)po/deixis]. Also Biese, Die Philosophie des Aristoteles, ch. iv. p. 249.]