Aristotle here again enforces what he had before urged--that in every valid syllogism, one premiss at least must be affirmative, and one premiss at least must be universal. If the conclusion be universal, both premisses must be so likewise; if it be particular, one of the premisses may not be universal. But without one universal premiss at least, there can be no syllogistic proof. If you have a thesis to support, you cannot assume (or ask to be conceded to you) that very thesis, without committing _petitio principii,_ (_i.e._ _quæsiti_ or _probandi_); you must assume (or ask to have conceded to you) some universal proposition containing it and more besides; under which universal you may bring the subject of your thesis as a minor, and thus the premisses necessary for supporting it will be completed. Aristotle illustrates this by giving a demonstration that the angles at the base of an isosceles triangle are equal; justifying every step in the reasoning by an appeal to some universal proposition.[42] [Footnote 42: Analyt. Prior. I. xxiv. p. 41, b. 6-31. The demonstration given (b. 13-22) is different from that which we read in Euclid, and is not easy to follow. It is more clearly explained by Waitz (p. 434) than either by Julius Pacius or by M. Barth. St. Hilaire (p. 108).] Again, every demonstration is effected by two propositions (an _even_ number) and by three terms (an _odd_ number); though the same proposition may perhaps be demonstrable by more than one pair of premisses, or through more than one middle term;[43] that is, by two or more distinct syllogisms. If there be more than three terms and two propositions, either the syllogism will no longer be one but several; or there must be particulars introduced for the purpose of obtaining an universal by induction; or something will be included, superfluous and not essential to the demonstration, perhaps for the purpose of concealing from the respondent the real inference meant.[44] In the case (afterwards called _Sorites_) where the ultimate conclusion is obtained through several mean terms in continuous series, the number of terms will always exceed by one the number of propositions; but the numbers may be odd or even, according to circumstances. As terms are added, the total of intermediate conclusions, if drawn out in form, will come to be far greater than that of the terms or propositions, multiplying as it will do in an increasing ratio to them.[45] [Footnote 43: Ibid. I. xxv. p. 41, b. 36, seq.] [Footnote 44: Ibid. xxv. p. 42, a. 23: [Greek: ma/tên e)/stai ei)lêmme/na, ei) mê\ e)pagôgê=s ê)\ kru/pseôs ê)/ tinos a)/llou tô=n toiou/tôn cha/rin.] Ib. a. 38: [Greek: ou(=tos o( lo/gos ê)\ ou) sullelo/gistai ê)\ plei/ô tô=n a)nagkai/ôn ê)rô/têke pro\s tê\n the/sin.]] [Footnote 45: Ibid. p. 42, b. 5-26.]