5. Prove that the number of undistributed terms in the premisses cannot exceed those in the conclusion by more than one. 6. Prove that wherever the minor premiss is negative, the major must be universal. 7. Prove that wherever the minor term is distributed, the major premiss must be universal. 8. If the middle term be twice distributed, what mood and figure are possible? 9. If the major term of a syllogism be the predicate of the major premiss, what do we know about the minor premiss? 10. When the middle term is distributed in both premisses, what must be the quantity of the conclusion? 11. Prove that if the conclusion be universal, the middle term can only be distributed once in the premisses. 12. Show how it is sometimes possible to draw three different conclusions from the same premisses. CHAPTER XIX. 1. Convert the following propositions-- (1) If a man is wise, he is humble. (2) Where there is sincerity there is no affectation. (3) When night-dogs run, all sorts of deer are chased. (4) The nearer the Church, the further from God. (5) If there were no void, all would be solid. (6) Not to go on is sometimes to go back. 2. Express in a single proposition-- If he was divine, he was not covetous; and if he was covetous, he was not divine. 3. Exhibit the exact logical relation to one another of the following pairs of propositions-- (1) If the conclusion be false, the premisses are false. If the conclusion be true, the premisses are not necessarily true. (2) If one premiss be negative, the conclusion must be negative. If the conclusion be negative, one of the premisses must be negative. (3) The truth of the universal involves the truth of the particular. The falsity of the particular involves the falsity of the universal. (4) From the truth of the particular no conclusion follows as to the universal. From the falsity of the universal no conclusion follows as to the particular. (5) If the conclusion in the fourth figure be negative, the major premiss must be universal. If the major premiss in the fourth figure be particular, the conclusion must be affirmative. (6) If both premisses be affirmative, the conclusion must be affirmative. If the conclusion be negative, one of the premisses must be negative. 4. 'The Method of Agreement stands on the ground that whatever circumstance can be eliminated is not connected with the phenomenon by any law; the Method of Difference stands on the ground that whatever circumstance cannot be eliminated is connected with the phenomenon by a law.' Do these two principles imply one another? CHAPTERS XX-XXVIII. 1. Fill up the following enthymemes, and state the exact nature of the resulting syllogism-- (1) If Livy is a faultless historian, we must believe all that he tells us; but that it is impossible to do.