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]
[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.
[27] Aristot. Analyt. Post. I. viii. p. 75, b. 21-36. Compare Metaphys. Z. p. 1040, a. 1: δῆλον ὅτι οὐκ ἂν εἴη αὐτῶν (τῶν φθαρτῶν) οὔθ’ ὁρισμὸς οὔτ’ ἀπόδειξις. Also Biese, Die Philosophie des Aristoteles, ch. iv. p. 249.
For complete demonstration, it is not sufficient that the premisses be true, immediate, and undemonstrable; they must, furthermore, be essential and appropriate to the class in hand. Unless they be such, you cannot be said to know the conclusion absolutely; you know it only by accident. You can only know a conclusion when demonstrated from its own appropriate premisses; and you know it best when it is demonstrated from its highest premisses. It is sometimes difficult to determine whether we really know or not; for we fancy that we know, when we demonstrate from true and universal principia, without being aware whether they are, or are not, the principia appropriate to the case.[28] But these principia must always be assumed without demonstration — the class whose essential constituent properties are in question, the universal Axioms, and the Definition or meaning of the attributes to be demonstrated. If these definitions and axioms are not always formally enunciated, it is because we tacitly presume them to be already known and admitted by the learner.[29] He may indeed always refuse to grant them in express words, but they are such that he cannot help granting them by internal assent in his mind, to which every syllogism must address itself. When you assume a premiss without demonstrating it, though it be really demonstrable, this, if the learner is favourable and willing to grant it, is an assumption or Hypothesis, valid relatively to him alone, but not valid absolutely: if he is reluctant or adverse, it is a Postulate, which you claim whether he is satisfied or not.[30] The Definition by itself is not an hypothesis; for it neither affirms nor denies the existence of anything. The pupil must indeed understand the terms of it; but this alone is not an hypothesis, unless you call the fact that the pupil comes to learn, an hypothesis.[31] The Hypothesis or assumption is contained in the premisses, being that by which the reason of the conclusion comes to be true. Some object that the geometer makes a false hypothesis or assumption, when he declares a given line drawn to be straight, or to be a foot long, though it is neither one nor the other. But this objection has no pertinence, since the geometer does not derive his conclusions from what is true of the visible lines drawn before his eyes, but from what is true of the lines conceived in his own mind, and signified or illustrated by the visible diagrams.[32]
[28] Ibid. ix. p. 75, b. 37-p. 76, a. 30.
[29] Ibid. x. p. 76, a. 31-b. 22.
[30] Aristot. Analyt. Post. I. x. p. 76, b. 29-34: ἐὰν μὲν δοκοῦντα λαμβάνῃ τῷ μανθάνοντι, ὑποτίθεται, καὶ ἔστιν οὔχ ἁπλῶς ὑπόθεσις, ἀλλὰ πρὸς ἐκεῖνον μόνον, ἂν δὲ ἢ μηδεμίᾶς ἐνούσης δόξης ἢ καὶ ἐναντίας ἐνούσης λαμβάνῃ τὸ αὐτό, αἰτεῖται. καὶ τούτῳ διαφέρει ὑπόθεσις καὶ αἴτημα, &c. Themistius, Paraphras. p. 37, Spengel.
[31] Ibid. p. 76, b. 36: τοῦτο δ’ οὐχ ὑπόθεσις, εἰ μὴ καὶ τὸ ἀκούειν ὑπόθεσίν τις εἶναι φήσει. For the meaning of τὸ ἀκούειν, compare ὁ ἀκούων, infra, Analyt. Post. I. xxiv. p. 85, b. 22.
[32] Ibid. p. 77, a. 1: ὁ δὲ γεωμέτρης οὐδὲν συμπεραίνεται τῷ τήνδε εἶναι τὴν γραμμὴν ἣν αὐτὸς ἔφθεγκται, ἀλλὰ τὰ διὰ τούτων δηλούμενα.
Themistius, Paraphr. p. 37: ὥσπερ οὐδ’ οἱ γεωμέτραι κέχρηνται ταῖς γραμμαῖς ὑπὲρ ὧν διαλέγονται καὶ δεικνύουσιν, ἀλλ’ ἃς ἔχουσιν ἐν τῇ ψυχῇ, ὧν εἰσὶ σύμβολα αἱ γραφόμεναι.