If two ratios are equal, their duplicate ratios are equal; and conversely. then shall the duplicate ratio of A: B be equal to the duplicate ratio of C: D. Let X be a third proportional to A, B; that is, the duplicate ratio of A: B=the duplicate ratio of C: D. Conversely, let the duplicate ratio of A: B be equal to the duplicate ratio of C: D; PROOFS OF THE PROPOSITIONS OF BOOK V. DERIVED FROM THE GEOMETRICAL DEFINITION OF PROPORTION. Obs. The Propositions of Book V. are all theorems. PROPOSITION 1. Ratios which are equal to the same ratio are equal to one another. Let A B P: Q, and also C: D:: P: Q; then shall A: BC: D. For it is evident that two scales or arrangements of multiples which agree in every respect with a third scale, will agree with one another. PROPOSITION 2. If two ratios are equal, the antecedent of the second is greater than, equal to, or less than its consequent according as the antecedent of the first is greater than, equal to, or less than its consequent. This follows at once from Def. 4, by taking m and n each equal to unity. PROPOSITION 3. If two ratios are equal, their reciprocal ratios are equal. For, by hypothesis, the multiples of A are distributed among those of B in the same manner as the multiples of C are among those of D; therefore also, the multiples of B are distributed among those of A in the same manner as the multiples of D are among those of C. NOTE. This proposition is sometimes enunciated thus If four magnitudes are proportionals, they are also proportionals when taken inversely, and the inference is referred to as invertendo or inversely. PROPOSITION 4. Equal magnitudes have the same ratio to the same magnitude; and the same magnitude has the same ratio to equal magnitudes. Let A, B, C be three magnitudes of the same kind, and let A be equal to B; Since AB, their multiples are identical and therefore are distributed in the same way among the multiples of C. .. AC: B: C, .. also, invertendo, CAC: B. Def. 4. v. 3. PROPOSITION 5. Of two unequal magnitudes, the greater has a greater ratio to a third magnitude than the less has; and the same magnitude has a greater ratio to the less of two magnitudes than it has to the greater. Since A > B, it will be possible to find m such that mA exceeds mB by a magnitude greater than C; hence if mA lies between nC and (n + 1)C, mB <nC: and if mA = nC, then mB <n©; For taking m and n as before, nc> mB, while nC is not > MĀ; .. C: B> CA. Def. 6. Def. 6. PROPOSITION 6. Magnitudes which have the same ratio to the same magnitude are equal to one another; and those to which the same magnitude has the same ratio are equal to one another. |