which is not a syllogism, since there is no middle term. Evident as the preceding is, the following additional proposition must be formed before it can be made explicitly logical. ‘If B be a magnitude less than C, then every magnitude less than B is also less than C.’ There is, then, before the preceding can be reduced to a syllogistic form, the necessity of a deduction from the second premiss, and the substitution of the result instead of that premiss. Thus, A is less than B Less than B is less than C: following from B is less than C. ————————— ————————————————— Therefore A is less than C But, if the additional argument be examined—namely, if B be less than C, then that which is less than B is less than C—it will be found to require precisely the same considerations repeated; for the original inference was nothing more. In fact, it may easily be seen as follows, that the proposition before us involves more than any simple syllogism can express. When we say that A is less than B, we say that if A were applied to B, every part of A would match a part of B, and there would be parts of B remaining over. But when we say, ‘Every A is B,’ meaning the premiss of a common syllogism, we say that every instance of A is an instance of B, without saying any thing as to whether there are or are not instances of B still left, after those which are also A are taken away. If, then, we wish to write an ordinary syllogism in a manner which shall correspond with ‘A is less than B, B is less than C, therefore A is less than C,’ we must introduce a more definite amount of assertion than was made in the preceding forms. Thus, Every A is B, and there are Bs which are not As Every B is C, and there are Cs which are not Bs ——————————————————————————————————————————————— Therefore Every A is C, and there are Cs which are not As Or thus: The Bs contain all the As, and more The Cs contain all the Bs, and more ——————————————————————————————————— The Cs contain all the As, and more The most technical form, however, is, From Every A is B; [Some B is not A] Every B is C; [Some C is not B] Follows Every A is C; [Some C is not A]