[Footnote 54: Analyt. Prior. II. xxiii. p. 68, b. 27: [Greek: dei= de\ noei=n to\ G to\ e)x a(pa/ntôn tô=n kath' e(/kaston sugkei/menon; ê( ga\r e)pagôgê\ dia\ pa/ntôn.]] [Footnote 55: Analyt. Prior. II. **xxiii. p. 68, p. 23: [Greek: ei) ou)=n a)ntistre/phei to\ G tô=| B, kai\ mê\ u(pertei/nei to\ me/son, a)na/gkê to\ A tô=| B u(pa/rchein.] Julius Pacius translates this: "Si igitur convertatur [Greek: to\ G] cum B, nec medium excedat, necesse est [Greek: to\ A tô=| B] inesse." These Latin words include the same grammatical ambiguity as is found in the Greek original: _medium_, like [Greek: to\ me/son], may be either an accusative case governed by _excedat_, or a nominative case preceding _excedat_. The same may be said of the other Latin translations, from Boethius downwards. But M. Barthélemy St. Hilaire in his French translation, and Sir W. Hamilton in his English translation (Lectures on Logic, Vol. II. iv. p. 358, Appendix), steer clear of this ambiguity. The former says: "Si donc C est réciproque à B, et qu'il ne dépasse pas le moyen, il est nécessaire alors que A soit à B:" to the same purpose, Hamilton, _l. c._ These words are quite plain and unequivocal. Yet I do not think that they convey the meaning of Aristotle. In my judgment, Aristotle meant to say: "If then C reciprocates with B, and if the middle term (B) does not stretch beyond (the minor C), it is necessary that A should be predicable of B." To show that this must be the meaning, we have only to reflect on what C and B respectively designate. It is assumed that C designates the sum of individual bile-less animals; and that B designates the class or class-term bile-less, that is, the totality thereof. Now the sum of individuals included in the minor (C) cannot upon any supposition overpass the totality: but it may very possibly fall short of totality; or (to state the same thing in other words) the totality may possibly surpass the sum of individuals under survey, but it cannot possibly fall short thereof. B is here the limit, and may possibly stretch beyond C; but cannot stretch beyond B. Hence I contend that the translations, both by M. Barthélemy St. Hilaire and Sir W. Hamilton, take the wrong side in the grammatical alternative admissible under the words [Greek: kai\ mê\ u(pertei/nei to\ me/son]. The only doubt that could possibly arise in the case was, whether the aggregate of individuals designated by the minor did, or did not, reach up to the totality designated by the middle term; or (changing the phrase) whether the totality designated by the middle term did, or did not, stretch beyond the aggregate of individuals designated by the minor. Aristotle terminates this doubt by the words: "And if the middle term does _not_ stretch beyond (the minor)." Of course the middle term does not stretch beyond, when the terms reciprocate; but when they do not reciprocate, the middle term must be the _more_ extensive of the