7. If there be two analogisms, where four antecedents make an analogism, their consequents also shall make an analogism; as also the sums of their antecedents will be proportional to the sums of their consequents. For if A. B :: C. D and E. F :: G. H be two analogisms, and A. E :: C. G be proportionals, then by permutation A. C :: E. G, and E. G :: F. H, and A. C :: B. D will be proportionals; wherefore B. D :: E. G, that is, B. D :: F. H, and by permutation B. F :: D. H are proportionals; which is the first. Secondly, I say A + E. B + F :: C + G. D + H will be proportionals. For seeing A. E :: C. G are proportionals, A + E. E :: C + G. G will also by composition be proportionals, and by permutation A + E. C + G :: E. G will be proportionals; wherefore, also A + E. C + G :: B + F. D + H will be proportionals. Again, seeing, as is shown above, B. F :: D. H are proportionals, B + F. F :: D + H. H will also by composition be proportionals; and by permutation B + F. D + H :: F. H will also be proportionals; wherefore A + E. C + G :: B + F. D + H are proportionals; which remained to be proved. Coroll. By the same reason, if there be never so many analogisms, and the antecedents be proportional to the antecedents, it may be demonstrated also that the consequents will be proportional to the consequents, as also the sum of the antecedents to the sum of the consequents. [Sidenote: The definition of hyperlogism and hypologism, that is, of greater and less proportion, and their transmutations.]