M. Barthélemy St. Hilaire remarks in the note to his translation of the Analytica Priora (p. 178): "Ce chapitre suffit à prouver qu'Aristote a distingué très-nettement les syllogismes par l'absurde, des syllogismes hypothétiques. Cette dernière dénomination est tout à fait pour lui ce qu'elle est pour nous." Of these two statements, I think the _latter_ is more than we can venture to affirm, considering that the general survey of hypothetical syllogisms, which Aristotle intended to draw up, either never was really completed, or at least has perished: the _former_ appears to me incorrect. Aristotle decidedly reckons the _Reductio ad Impossibile_ among hypothetical proofs. But he understands by _Reductio ad Impossibile_ something rather wider than what the moderns understand by it. It now means only, that you take the contradictory of the conclusion together with one of the premisses, and by means of these two demonstrate a conclusion contradictory or contrary to the other premiss. But Aristotle understood by it this, and something more besides, namely, whenever, by taking the contradictory of the conclusion, together with some other incontestable premiss, you demonstrate, by means of the two, some new conclusion notoriously false. What I here say, is illustrated by the very example which he gives in this chapter. The incommensurability of the diagonal (with the side of the square) is demonstrated by _Reductio ad Impossibile_; because if it be supposed commensurable, you may demonstrate that an odd number is equal to an even number; a conclusion which every one will declare to be inadmissible, but which is not the contradictory of either of the premisses whereby the true proposition was demonstrated.] Here Aristotle expressly reserves for separate treatment the general subject of Syllogisms from Hypothesis.[88] [Footnote 88: The expressions of Aristotle here are remarkable, Analyt. Prior. I. xliv. p. 50, a. 39-b. 3: [Greek: polloi\ de\ kai\ e(/teroi perai/nontai e)x u(pothe/seôs, ou(\s e)piske/psasthai dei= kai\ diasêmê=nai katharô=s. ti/nes me\n ou)=n ai( diaphorai\ tou/tôn, kai\ posachô=s gi/netai to\ e)x u(pothe/seôs, u(/steron e)rou=men; nu=n de\ tosou=nton ê(mi=n e)/stô phanero/n, o(/ti ou)k e)/stin a)nalu/ein ei)s ta\ schê/mata tou\s toiou/tous sullogismou/s. kai\ di' ê(\n ai)ti/an, ei)rê/kamen.] Syllogisms from Hypothesis were many and various, and Aristotle intended to treat them in a future treatise; but all that concerns the present treatise, in his opinion, is, to show that none of them can be reduced to the three Figures. Among the Syllogisms from Hypothesis, two varieties recognized by Aristotle (besides [Greek: oi) dia\ tou= a)duna/tou]) were [Greek: oi( kata\ meta/lêpsin] and [Greek: oi( kata\ poio/têta]. The same proposition which Aristotle entitles [Greek: kata\ meta/lêpsin], was afterwards designated by the Stoics [Greek: kata\ pro/slêpsin] (Alexander ap. Schol. p. 178, b. 6-24).