But pmD contains E as often as pmA contains B; and pnC contains B as often as pnF contains E. Therefore pmD contains E oftener than pnF contains E. Consequently mD > nF. In like manner may it be proved if nC > mA that nF> mD. Lastly, let mA=nC then (prop. ii.) E: mD :: E : nF. But, in this proportion, the antecedents are equal; therefore (prop. x. and vii.) mD=nF. It has now been proved that mA, mD being any equimultiples of A, D; and nC, nF any equimultiples of C, F; if mA is greater than nC, equal to it, or less; so will mD be greater than nF, equal to it, or less. Hence (def. v., p. 148.) A C D : F. Next, let there be four magnitudes in each series, viz. A, B, C, K; and D, E, F, L, furnishing the additional proportion CK FL, then taking this in conjunction with the conclusion just obtained, we see that the two series of magnitudes A, C, K and D, F, L are related to each other as the two series in the first case. Hence, as before, and in like manner might we proceed from the case of four magnitudes to that of five in each series; and so on, to any number.* COR. It follows from the above that if the consequents in one proportion be the antecedents in another, a third proportion may be formed, having the same antecedents as the first proportion, and the same consequents as the second. * This proposition is said to be inferred "ex æquali," or "from equality of distance." P If there be three magnitudes, and other three such that whichever set be taken, the first is to the second in that set as the second to the third in the other set; then if the first in one set be greater than the third, the first in the other set will be greater than the third. Let the one set of magnitudes be A, B, C, and the other set D, E, F; which are such that A: B:: E: F, and also D: E:: B: C. Let also the first in one set, as A, be greater than the third C; then also D > F. Take mA, mC such equimultiples of A, C that mA may contain B oftener than mC contains B. Then since mC contains B as often as mE contains D, therefore mA contains B oftener than mE contains D. But mE contains F as often as mA contains B. Therefore mE contains F oftener than mE contains D; and consequently D > F. In like manner may it be proved that. if D > F, then A > C. PROP. XVIII. THEOR. If there be three magnitudes, and other three such that whichever set be taken, the first is to the second in that set as the second to the third in the other set; then the first will be to the third in one set as the first to the third in the other. Let one set of magnitudes be A, B, C, and the other set D, E, F; so related that A: B:: E: F, and D:E::B:C; then also A: C:: D: F. Take of A, B, D any equimultiples mA, mB, mD; and of C, E, F any equimultiples nC, nE, nF. Then (prop. v., cor. iii.) and (prop. vi.) mA mB: nE: nF, mDnE: mB: nC. Hence the three magnitudes mA, mB, nC are related to other three mD, nE, nF as in prop. xvii. Consequently if mA > nC, then mD > nF, or if mD > nF, then mA > nC. But mA, mD are any equimultiples of A, D; and nC, nF any equimultiples of C, F; therefore (prop. iv.) If to one of the sets a fourth magnitude P be annexed, and to the other set a magnitude Q be prefixed such as to furnish the additional proportion Q: D:: C: P, then it may be easily proved from the above that in these sets of four, viz. A, B, C, P, and Q, D, E, F, the first is to the last in the one, as the first to the last in the other. For this proportion in conjunction with that deduced above shows that the three magnitudes A, C, P, and the other three Q, D, F are related as in the proposition; therefore APQ F. PROP. XIX. THEOR. If four magnitudes be proportional, the sum of the first two is to their difference as the sum of the other two is to their difference. Let A B C : D; then (prop. xiii.) : ABA: C + D : C, and (prop. xiv.) A: A B C C~ D. * This proportion is said to be inferred “ex æquali in proportione perturbata," or "from equality in perturbed or disorderly proportion." Consequently (prop. xvi. cor.) A + B A B C + D C ~ D. PROP. XX. THEOR. If the antecedents in one proportion be the same as those in another, then the first antecedent is to the sum or difference of the first consequents as the second antecedent is to the sum or difference of the second consequents. Let the proportions be A: B:: CD and A: E:: C: F, then also AB±E::C: D ± F. For inverting the first proportion BAD: C, and since A EC: F; BE D F, therefore (prop. xvi. cor.) and consequently (prop. xiii. and xiv.) B: BED: DF, hence (prop. xvi.cor.) comparing this with the first proportion A BEC: DF. COR. If the terms are all HOMOGENEOUS, then, by alternation, THE ELEMENTS OF EUCLID. BOOK VI. DEFINITIONS. I. Similar rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportional. II. Two sides of one figure are said to be reciprocally proportional to two sides of another, when one of the sides of the first is to one of the sides of the second, as the remaining side of the second to the remaining side of the first. III. Astraight line is said to be cut in extreme and mean proportion, when the whole is to the greater segment, as the greater segment is to the less. IV. The altitude of any figure is the straight line drawn from its vertex perpendicular to the base. P 2 |