To answer this question we have to consider whether the Conclusion can be drawn from either of the two premisses without the help of the other. If it is possible immediately, it must be educible directly either from the Major Premiss or from the Minor.
(a) Some logicians argue as if the Conclusion were immediately possible from the Major Premiss. The Minor Premiss and the Conclusion, they urge, are simply equivalent to the Major Premiss. But this is a misunderstanding. "If A is B, C is D," is not equivalent to "A is B, therefore C is D". "If the harbour is frozen, the ships cannot come in" is not to say that "the harbour is frozen, and therefore," etc. The Major Premiss merely affirms the existence of the relation of Reason and Consequent between the two propositions. But we cannot thereupon assert the Conclusion unless the Minor Premiss is also conceded; that is, the inference of the Conclusion is Mediate, as being from two premisses and not from one alone.
(b) Similarly with Hamilton's contention that the Conclusion is inferrible immediately from the Minor Premiss, inasmuch as the Consequent is involved in the Reason. True, the Consequent is involved in the Reason: but we cannot infer from "A is B" to "C is D," unless it is conceded that the relation of Reason and Consequent holds between them; that is, unless the Major Premiss is conceded as well as the Minor.
(3) Can Hypothetical Syllogism be reduced to the Categorical Form?
To oppose Hypothetical Syllogisms to Categorical is misleading, unless we take note of the precise difference between them. It is only in the form of the Major Premiss that they differ: Minor Premiss and Conclusion are categorical in both. And the meaning of a Hypothetical Major Premiss (unless it is a mere arbitrary convention between two disputants, to the effect that the Consequent will be admitted if the Antecedent is proved, or that the Antecedent will be relinquished if the Consequent is disproved), can always be put in the form of a general proposition, from which, with the Minor Premiss as applying proposition, a conclusion identical with the original can be drawn in regular Categorical form.
Thus:—
If the harbour is frozen, the ships cannot come in.
The harbour is frozen.
... The ships cannot come in.
This is a Hypothetical Syllogism, Modus Ponens. Express the Hypothetical Major in the form of the general proposition which it implies, and you reach a conclusion (in Barbara) which is only grammatically different from the original.