[Footnote 54: Analyt. Post. I. xviii. p. 81, a. 38-b. 9. In this important chapter (the doctrines of which are more fully expanded in the last chapter of the Second Book of the Analyt. Post.), the text of Waitz does not fully agree with that of Julius Pacius. In Firmin Didot's edition the text is the same as in Waitz; but his Latin translation remains adapted to that of Julius Pacius. Waitz gives the substance of the chapter as follows (ad Organ. II. p. 347):-- "Universales propositiones omnes inductione comparantur, quum etiam in iis, quæ a sensibus maxime aliena videntur et quæ, ut mathematica ([Greek: ta\ e)x a)phaire/seôs]), cogitatione separantur à materia quacum conjuncta sunt, inductione probentur ea quæ de genero (e. g., de linea vel de corpore mathematico), ad quod demonstratio pertineat, prædicentur [Greek: kath' au(ta/] et cum ejus natura conjuncta sint. Inductio autem iis nititur quæ sensibus percipiuntur; nam res singulares sentiuntur, scientia vero rerum singularium non datur sine inductione, non datur inductio sine sensu."] Aristotle next proceeds to show (what in previous passages he had assumed)[55] that, if Demonstration or the syllogistic process be possible--if there be any truths supposed demonstrable, this implies that there must be primary or ultimate truths. It has been explained that the constituent elements assumed in the Syllogism are three terms and two propositions or premisses; in the major premiss, A is affirmed (or denied) of all B; in the minor, B is affirmed of all C; in the conclusion, A is affirmed (or denied) of all C.[56] Now it is possible that there may be some one or more predicates higher than A, but it is impossible that there can be an infinite series of such higher predicates. So also there may be one or more subjects lower than C, and of which C will be the predicate; but it is impossible that there can be an infinite series of such lower subjects. In like manner there may perhaps be one or more middle terms between A and B, and between B and C; but it is impossible that there can be an infinite series of such intervening middle terms. There must be a limit to the series ascending, descending, or intervening.[57] These remarks have no application to reciprocating propositions, in which the predicate is co-extensive with the subject.[58] But they apply alike to demonstrations negative and affirmative, and alike to all the three figures of Syllogism.[59] [Footnote 55: Analyt. Prior. I. xxvii. p. 43, a. 38; Analyt. Post. I. ii. p. 71, b. 21.] [Footnote 56: Analyt. Post. I. xix. p. 81, b. 10-17.] [Footnote 57: Ibid. p. 81, b. 30-p. 82, a. 14.] [Footnote 58: Ibid. p. 82, a. 15-20. M. Barthélemy St. Hilaire, p. 117:--"Ceci ne saurait s'appliquer aux termes réciproques, parce que dans les termes qui peuvent être attribués réciproquement l'un à l'autre, on ne peut pas dire qu'il y ait ni premier ni dernier rélativement à l'attribution."] [Footnote 59: Analyt. Post. I. xx., xxi. p. 82, a. 21-b. 36.]