[Footnote 17: Dr. Wallis, in one of his acute controversial treatises against Hobbes, remarks upon this as the process pursued by Euclid in his demonstrations:--"You tell us next that an Induction, without enumeration of all the particulars, is not sufficient to infer a conclusion. Yes, Sir, if after the enumeration of some particulars, there comes a general clause, _and the like in other cases_ (as here it doth), this may pass for a proofe till there be a possibility of giving some instance to the contrary, which here you will never be able to doe. And if such an Induction may not pass for proofe, there is never a proposition in Euclid demonstrated. For all along he takes no other course, or at least grounds his Demonstrations on Propositions no otherwise demonstrated. As, for instance, he proposeth it in general (i. c. 1.)--_To make an equilateral triangle on a line given_. And then he shows you how to do it upon the line A B, which he there shows you, and leaves you to supply: _And the same, by the like means, may be done upon any other strait line_; and then infers his general conclusion. Yet I have not heard any man object that the Induction was not sufficient, because he did not actually performe it in all lines possible."--(Wallis, Due Correction to Mr. Hobbes, Oxon. 1656, sect. v. p. 42.) This is induction by _parity of reasoning_. So also Aristot. Analyt. Poster. I. iv. p. 73, b. 32: [Greek: to\ katho/lou de\ u(pa/rchei to/te, o(/tan e)pi\ tou= tucho/ntos kai\ prô/tou deiknu/êtai.]] Aristotle passes next to Affirmatives, both Universal and Particular. First, if A can be predicated of all B, then B can be predicated of _some_ A; for if B cannot be predicated of any A, then (by the rule for the Universal Negative) neither can A be predicated of any B. Again, if A can be predicated of some B, in this case also, and for the same reason, B can be predicated of some A.[18] Here the rule for the Universal Negative, supposed already established, is applied legitimately to prove the rules for Affirmatives. But in the first case, that of the Universal, it fails to prove _some_ in the sense of _not-all_ or _some-at-most_, which is required; whereas, the rules for both cases can be proved by Induction, like the formula about the Universal Negative. When we come to the Particular Negative, Aristotle lays down the position, that it does not admit of being necessarily converted in any way. He gives no proof of this, beyond one single exemplification: If some animal is not a man, you are not thereby warranted in asserting the converse, that some man is not an animal.[19] It is plain that such an exemplification is only an appeal to Induction: you produce one particular example, which is entering on the track of Induction; and one example alone is sufficient to establish the negative of an universal proposition.[20] The converse of a Particular Negative is not in all cases true, though it may be true in many cases.