PROP. IV. 4. THEOREM. If the first of four proportional magnitudes be a multiple, or a part, of the second, the third is the same multiple, or the same part, of the fourth. If A B C D, and if A=pB, then C-pD. : : For (hyp. and Supp. iii. 5.) A : B :: pD : D; and (hyp.) A B C D; therefore (E. xi. 5.) C: D:: pD: D; therefore (E. ix. 5.) C=pD. Again, if A: B :: C: D, and if pA=B, then PC=D. For (hyp.) A B C D; .. (Supp. ii. 5.) B: A: D: C; and (hyp.) B=pA; therefore, as in the former case, D=pC; that is, C is the same part of D, that A is of B. PROP. V. 5. THEOREM. If any number of equal ratios be each greater than a given ratio, the ratio of the sum of their antecedents to the sum of their consequents, shall be greater than that given ratio. Let the ratios (A ; B), (C : D), (E : F), &c. be equal to one another, and let each of them be greater than the ratio (P: Q); then (4+ C+E : B+D+F) > (P: Q.) For (E. xii. 5.) A+ C+E : B+D+F :: A : B; and (hyp.) (A: B) > (P : Q); :. (A + C+E : B+D+F) > P : Q. 6. THEOREM. If the first of four magnitudes have a greater ratio to the second than the third has to the fourth, the second shall have to the first a less ratio than the fourth has to the third. If (A : B) > (C : D), then is (B : A) < (D : C). and since (hyp.) (E : B) :: (C : D); (A : B) > (C : D) ; .. (A : B) > (E : B); therefore (E. viii. 5.) (B: E) > (B: A): But (hyp. and Supp. ii. 5.) (D: C) :: (B: E); therefore (E. xiii. 5.) (D : C) > (B : A): Or, (B: A) < (D : C). PROP. VII. 7. THEOREM. If the first of four magnitudes, of the same kind, have a greater ratio to the second than the third has to the fourth, the first shall have to the third a greater ratio than the second has to the fourth. If (A B) be greater than (C: D), then is (A : C) > (B : D). For, let E be a magnitude such that (E: B): (C: D); therefore (hyp. and E. x. 5.) A > E; therefore (E. viii. 5.) (A : C) > (E : C'); But (E. xvi. 5. and hyp.) (E: C) :: (B: D); .. (A : C) > (B : D). PROP. VIII. 8. THEOREM. If four magnitudes of the same kind be proportionals, and if the first of them be the greatest, the fourth shall be the least; but if the first of them be the least, the fourth shall be the greatest. Let A, B, C, D, be four magnitudes of the same kind, which are proportionals; and, first, let A be the greatest; then D shall be the least of them. For, since (hyp.) A > C; therefore (E. xiv. 5.) B>D; : :: C D; Again, since (hyp.) A B therefore (E. xvi. 5.) A: C:: B : D: But (hyp.) A > B; therefore (E. xiv. 5.) C > D: And it has been shewn that B > D; therefore D is in this case the least of the four proportionals. And, if A be the least of the four proportionals, it may, in like manner, be proved that D will be the greatest of them. 9. COR. If four magnitudes of the same kind, be continual proportionals, the difference between the two extremes is greater than the difference between the two means.. PROP. IX. 10. THEOREM. If the first, together with the second, of four magnitudes, have a greater ratio to the second, than the third, together with the fourth, has to the fourth, the first shall have a greater ratio to the second than the third has to the fourth. If (A + B : B) > (C+D then is (AB) > (C' : D). For, let E be a magnitude such that (E+B : B) :: (C+D : D); therefore (E. x. 5.) A + B > E+B; .. A > E; therefore (E. viii. 5.) (A : B) > (E : B) : But (hyp. and E. xvii. 5.) (E: B)=(C : D); .. (A : B) > (C : D). PROP. X. 11. THEOREM. If the first of four magnitudes have a greater ratio to the second than the third has to the fourth, the first, together with the second, shall have to the second, a greater ratio than the third, together with the fourth, has to the fourth. If (A : B) > (C : D); then is (A+B : B) > (C + D : D). For, let E be a magnitude such that (E: B) :: (C: D); therefore (E. x. 5.) A > E; .. A + B > E+B; therefore (E. viii. 5.) (A+B : B) > (E + B : B): But (E. xviii. 5. and hyp.) (E+B: B) :: (C+D: D); therefore (E. xi. 5.) (A+B : B) > (C+D : D). PROP. XI. 12. THEOREM. If the first term of a ratio be less than the second, the ratio shall be increased by adding the same quantity to both terms; but if the first term be greater than the second, the ratio shall be diminished by adding the same quantity to both. Let A < B, and let C be any other magnitude: Then is (4+C : B+C) > (A : B). For, (E. xviii. 5. and hyp.), (C : A) >(C : B); therefore (Supp. x. 5.), (4+C: A) > (B+C: B); therefore (Supp. vii. 5.) (A+C: B+C)>(A : B). And, if A be greater than B, it may, in the same manner, be shewn that (A+ C : B+C) > (A : B). 13. COR. If a< A and b< B, a fourth proportional to A+B, A and B, shall be greater than a fourth proportional to a+b, a and b. For, let A+b : A :: b : D, and a+b: a :: b : d. And, since (12) (a+b: a)>(A+b: A) D>d. In like manner, if A+B : B :: A: E, it may be shewn that E>D: Much more, then, is E>d. PROP. XII. 14. THEOREM. If the first of four magnitudes, of the same kind, have a greater ratio to the second than the third has to the fourth, the first, together with the third, shall have to the second, together with the fourth, a greater ratio than the third has to the fourth, and a less ratio than the first has to the second. If (A: B) be greater than (C: D), then is (A+C: B+D) > (C : D) ; and (4+C: B+D) < (A : B). For, (Supp. vii. 5. and hyp.) (A: C) >(A : D); therefore (Supp. x. 5.), (A+C: C) > (B+D: D); therefore (Supp. vii. 5.), (A+ C : B+D) > (C: D): Again, since (hyp. and Supp. vi. 5.), (B : A) < (D: C), or (D: C)>(B: A), it may be shewn, in the same manner, that (A+C : B+D) < A : B. |