tionals, they are proportionals also when taken inversely. Let A:B:: C:D; then B: A:: D: C. Let there be taken of A and C any equi-multiples pA, PC, and of B. and D any equi-multiples qB, qD: Then (hyp.) A: B:: C: D, ... (5. def. 5.) if pA>qB, pC>qD; if pA = qB, pC=qD; if pA <qB, pC<qD; .. accordingly as qB is greater than, equal to, or less than pA, qD is greater than, equal to, or less than pC; .. (5. def. 5.) B: A:: D: C. PROP. III. 3. THEOREM. If the first of four magnitudes be the same multiple of the second, or the same part of it, that the third is of the fourth, the first is to the second, as the third is to the fourth. First, let A = pB, and C = pD; then A:B:: C: D. Let there be taken of A and C any equi-multiples qA, qC, and of B and D any equi-multiples rB, rD. And since A = pB, accordingly as qà >,=, or < rB, will q times pB be >,=, or < rB, i. e. q times p will be >,, or <r; and .. q times pD will also be >,, or <rD; i. e. (hyp.) qC will be >,=, or < rD; ... (5. def. 5.) A:B::C:D. Secondly, let pA=B, and pC=D; then, also, A:B::C: D. For in that case, as hath been shewn, B:A::D:C; .. (S. 2. 5.) A: B:: C: D. 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 S. 3. 5.) A: B:: pD:D; and (hyp.) A: B:: C:D; .. (E. 11. 5.) C: D::pD:D; .. (E. 9. 5.) C=pD. Again, if A: B:: C: D, and if pA B, then pC=D. For (hyp.) A: B::C: D; .. (S. 2. 5.) B: A:: D:C; and (hyp.) BpA; .., as in the former case, DpC; i. e. 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 (A + C+E :B+D+F)>(P: Q.) For (E. 12. 5.) A+C+E:B+D+F::A:B; and (hyp.) (A: B)>(P: Q); ··. (A+C+E: B+D+F)>P: Q. PROP. VI. 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). For, let E be a magnitude such that (E: B):: (C: D); and since (hyp.) (A: B) > (C: D) .·., (A : B) > (E: B); ... (E. 8. 5.) (B: E)>(B: A); 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 .. (hyp. and E. 10. 5.) A > E But (E. 16. 5. and hyp.) (E: 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, .. (E. 14.5.) B>D; .. (E. 16. 5.) A: C:: B: D: But (hyp.) A>B; .'. (E. 14. 5.) C>D: And it has been shewn that B >D; .. 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 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 |