[Footnote 12: Aristot. Analyt. Post I. ii. p. 71, b. 9-17. Julius Pacius says in a note, ad c. ii. p. 394: "Propositio demonstrativa est prima, immediata, et indemonstrabilis. His tribus verbis significatur una et eadem conditio; nam propositio prima est, quæ, quod medio caret, demonstrari nequit." So also Zabarella (In lib. I. Post. Anal. Comm., p. 340, Op. ed. Venet. 1617): "Duæ illæ dictiones (_primis_ et _immediatis_) unam tantum significant conditionem ordine secundam, non duas; idem namque est, principia esse medio carentia, ac esse prima."] [Footnote 13: Aristot. Analyt. Post. I. ii. p. 72, a. 1-24; Themistius, Paraphr. I. ii. p. 10, ed. Spengel; Schol. p. 199, b. 44. Themistius quotes the definition of an Axiom as given by Theophrastus: [Greek: A)xi/ôma/ e)sti _do/xa_ tis], &c. This shows the difficulty of adhering precisely to a scientific terminology. Theophrastus explains an axiom to be a sort of [Greek: do/xa], thus lapsing into the common loose use of the word. Yet still both he and Aristotle declare [Greek: do/xa] to be of inferior intellectual worth as compared with [Greek: e)pistê/mê] (Anal. Post. I. xxiii.), while at the same time they declare the Axiom to be the very maximum of scientific truth. Theophrastus gave, as examples of Axioms, the **maxim of Contradiction, universally applicable, and, "If equals be taken from equals the remainders will be equal," applicable to homogeneous quantities. Even Aristotle himself sometimes falls into the same vague employment of [Greek: do/xa], as including the Axioms. See Metaphys. B. ii. p. 996, b. 28; [Greek: G]. iii. p. 1005, b. 33.] [Footnote 14: Aristot. Anal. Post. I. ii. p. 72, a. 25, b. 4. I translate these words in conformity with Themistius, pp. 12-13, and with Mr. Poste's translation, p. 43. Julius Pacius and M. Barthélemy St. Hilaire render them somewhat differently. They also read [Greek: a)meta/ptôtos], while Waitz and Firmin Didot read [Greek: a)meta/peistos], which last seems preferable.] In Aristotle's time two doctrines had been advanced, in opposition to the preceding theory: (1) Some denied the necessity of any indemonstrable _principia_, and affirmed the possibility of, demonstrating backwards _ad infinitum_; (2) Others agreed in denying the necessity of any indemonstrable _principia_, but contended that demonstration in a circle is valid and legitimate--_e.g._ that A may be demonstrated by means of B, and B by means of A. Against both these doctrines Aristotle enters his protest. The first of them--the supposition of an interminable regress--he pronounces to be obviously absurd: the second he declares tantamount to proving a thing by itself; the circular demonstration, besides, having been shown to be impossible, except in the First figure, with propositions in which the predicate reciprocates or is co-extensive with the subject--a very small proportion among propositions generally used in demonstrating.[15]