Distinct from all these three, however, Aristotle singles out and dwells upon another mode of error, which he calls _Petitio Principii_. Some truths, the _principia_, are by nature knowable through or in themselves, others are knowable only through other things. If you confound this distinction, and ask or assume something of the latter class as if it belonged to the former, you commit a _Petitio Principii_. You may commit it either by assuming at once that which ought to be demonstrated, or by assuming, as if it were a _principium_, something else among those matters which in natural propriety would be demonstrated by means of a _principium_. Thus, there is (let us suppose) a natural propriety that C shall be demonstrated through A; but you, overlooking this, demonstrate B through C, and A through B. By thus inverting the legitimate order, you do what is tantamount to demonstrating A through itself; for your demonstration will not hold unless you assume A at the beginning, in order to arrive at C. This is a mistake made not unfrequently, and especially by some who define parallel lines; for they give a definition which cannot be understood unless parallel lines be presupposed.[20] [Footnote 20: Analyt. Prior. II. xvi. p. 64, b. 33-p. 65, a. 9. _Petere principium_ is, in the phrase of Aristotle, not [Greek: tê\n a)rchê\n ai)tei=sthai], but [Greek: to\ e)n a)rchê=| ai)tei=sthai] or [Greek: to\ e)x a)rchê=s ai)tei=sthai] (xvi. p. 64, b. 28, 34).] When the problem is such, that it is uncertain whether A can be predicated either of C or of B, if you then assume that A is predicable of B, you may perhaps not commit _Petitio Principii_, but you certainly fail in demonstrating the problem; for no demonstration will hold where the premiss is equally uncertain with the conclusion. But if, besides, the case be such, that B is identical with C, that is, either co-extensive and reciprocally convertible with C, or related to C as genus or species,--in either of these cases you commit _Petitio Principii_ by assuming that A may be predicated of B.[21] For seeing that B reciprocates with C, you might just as well demonstrate that A is predicable of B, because it is predicable of C; that is, you might demonstrate the major premiss by means of the minor and the conclusion, as well as you can demonstrate the conclusion by means of the major and the minor premiss. If you cannot so demonstrate the major premiss, this is not because the structure of the syllogism forbids it, but because the predicate of the major premiss is more extensive than the subject thereof. If it be co-extensive and convertible with the subject, we shall have a circular proof of three propositions in which each may be alternately premiss and conclusion. The like will be the case, if the _Petitio Principii_ is in the minor premiss and not in the major. In the First syllogistic figure it may be in either of the premisses; in the