8. In an hyperlogism, that is, where the proportion of the first antecedent to its consequent is greater than the proportion of the second antecedent to its consequent, the permutation of the proportionals, and the addition of proportionals to proportionals, and substraction of them from one another, as also their composition and division, and their multiplication and division by the same number, produce always an hyperlogism. For suppose A. B :: C . D and A. C :: E. F be analogisms, A + E. B :: C + F . D will also be an analogism; but A + E. B :: C. D will be an hyperlogism; wherefore by permutation, A + E. C :: B. D is an hyperlogism, because A. B :: C. D is an analogism. Secondly, if to the hyperlogism A + E. B :: C. D the proportionals G and H be added, A + E + G. B :: C + H. D will be an hyperlogism, by reason A + E + G. B :: C + F + H. D is an analogism. Also, if G and H be taken away, A + E - G. B :: C - H. D will be an hyperlogism; for A + E - G. B :: C + F - H. D is an analogism. Thirdly, by composition A + E + B. B :: C + D. D will be an hyperlogism, because A + E + B. B :: C + F + D. D is ah analogism, and so it will be in all the varieties of composition. Fourthly, by division, A + E - B. B :: C - D. D will by an hyperlogism, by reason A E - B. B :: C + F - D. D is an analogism. Also A + E - B. A + E :: C - D. C is an hyperlogism; for A + E - B. A + E :: C + F - D. C is an analogism. Fifthly, by multiplication 4 A + 4 E. B :: 4 C. D is an hyperlogism, because 4 A. B :: 4 C. D is an analogism; and by division ¼ A + ¼ E. B :: ¼ C.D is an hyperlogism, because ¼ A. B :: ¼ C. D is an analogism. 9. But if A + E. B :: C. D be an hyperlogism, then by inversion B. A + E :: D. C will be an hypologism, because B. A :: D. C being an analogism, the first consequent will be too great. Also, by conversion of proportion, A + E. A + E - B :: C. C - D is an hypologism, because the inversion of it, namely A + E - B. A + E :: C - D. C is an hyperlogism, as I have shown but now. So also B. A + E - B :: D. C - D is an hypologism, because, as I have newly shown, the inversion of it, namely A + E - B. B :: C - D. D is an hyperlogism. Note that this hypologism A + E. A + E - B :: C. C - D is commonly thus expressed; if the proportion of the whole, (A + E) to that which is taken out of it (B), be greater than the proportion of the whole (C) to that which is taken out of it (D), then the proportion of the whole (A + E) to the remainder (A + E - B) will be less than the proportion of the whole (C) to the remainder (C - D). [Sidenote: Comparison of analogical quantities, according to magnitude.]