44 A SYLLABUS OF [Let A:B ::C:D, then A :C:: B : D. therefore mA : mB :: nC : nd, A:C :: B:D.] THEOR. 9. If any number of magnitudes of the same kind be proportionals, as one of the antecedents is to its consequent, so shall the sum of the antecedents be to the sum of the consequents. . [Let A : B :: C:D :: E : F, then A:B :: A+C+E : B+D+F. For as mA>=<nB, so is mC>=or<nd, and so also is mE>=or <nF; whence it follows that so also is mA +mC+mE>=or <nB +nD+nF and therefore so is m (A +C+E)>=or<n (B+D+F), whence A:B :: A+C+E : B+D+F.] THEOR. 10. If two ratios are equal, the sum or difference of the antecedent and consequent of the first has to the so is m (P+Q) between (m+n) and (m+n+I)Q or A+B : B :: P+Q: Q. may be proved that A~B :B :: P~Q : Q.] or COR. If two ratios are equal, the sum or difference of the antecedent and consequent of the first has to their difference or sum the same ratio as the sum difference of the antecedent and consequent of the other has to their difference or sum. THEOR. 11. If two ratios are equal, and equimultiples of the antecedents and also of the consequents are taken, the multiple of the first antecedent has to that of its consequent the same ratio as the multiple of the other antecedent has to that of its consequent. (Let A :B :: P: Q, then mA : nB :: mP: nQ. For pm.A>=or <qn.B, as pm.P>=or <an.Q, and therefore p.mA >=or <q nB, as p.mP >=or <q.nl, whence, p, 2 being any numbers whatever, MA : nB :: mP : nQ.] THEOR. 12. If there be two sets of magnitudes, such that the first is to the second of the first set as the first to the second of the other set, and the second to the third of the first set as the second to the third of the other, and so on to the last magnitude: then the first is to the last of the first set as the first to the last of the other. [Let the two sets of three magnitudes be A, B, C and P, Q, R, and let A:B :: P : Q and B :C:: Q : R, then A: C :: P: R. 46 A SYLLABUS OF For if A>C, A : B>C : B and C :B :: R:Q, therefore P :Q>R: Q, whence P>R. A: C :: P: R. then, since A :B :: P : Q and B :C :: Q : R, therefore A :C :: P:R; but C :D :: R: S, and therefore A D :: P : S.] COR. If A:B::Q:R and B:C:: P:Q, then A:C :: P: R. [Let S be a fourth proportional to Q, R, P, then Q : R :: P : S, Hence A : B :: P : S and B : C :: S : R, DEF. 9. If there are any number of magnitudes of the same kind, the first is said to have to the last the ratio compounded of the ratios of the first to the second, of the second to the third, and so on to the last magnitude. DEF. 10. If there are any number of ratios, and a set of magnitudes is taken such that the ratio of the first to the second is equal to the first ratio, and the ratio of the second to the third is equal to the second ratio, and so on, then the first of the set is said to have to the last the ratio compounded of the original ratios. OBS. From these Definitions it follows, by Theor. 12, that if there be two sets of ratios equal to one another, each to each, the ratio compounded of the ratios of the first set is equal to that compounded of the ratios of the its reciprocal is the ratio of equality. Def. II. When two ratios are equal, the ratio compounded of them is called the duplicate ratio of either of the original ratios. DEF. 12. When three ratios are equal, the ratio compounded of them is called the triplicate ratio of any one of the original ratios. SECTION 2. FUNDAMENTAL GEOMETRICAL PROPOSITIONS. THEOR, I. If two straight lines are cut by three parallel straight lines, the intercepts on the one are to one another in the same ratio as the corresponding intercepts on the other. (Let the three parallel lines AA', BB', CC', cut other two lines in A, B, C, and A', B', C' respectively: then AB : BC :: A'B' : B'C'. On the line ABC take BM=m.AB and BN=n.BC, M and N being taken on the same side of B. Also on the line A'B'C' take B M'=m. A'B' and B'N'= n.B'C', M', N' being on the same side of B' as M, N are of B. It is easy to prove that MM and NN’ are both parallel to BB'. Hence, whatever be the values of 11 and n, as BM (or m. AB) is greater than, equal to, or less than BN (or n.BC), so is B M' (or m. A'B') greater than, equal to, or less than B'N' (or n.B'C'), therefore AB : BC :: A'B' : B'C'. It will be observed that the reasoning holds good, whether B be between A and C or beyond A or beyond C.] 48 A SYLLABUS OF COR. 1. If the sides of a triangle are cut by a straight line parallel to the base, the segments of one side are to one another in the same ratio as the segments of the other side. COR. 2. If two straight lines are cut by four parallel straight lines the intercepts on the one are to one another in the same ratio as the corresponding intercepts on the other. THEOR. 2. A given finite straight line can be divided internally into segments having any given ratio, and also externally into segments having any given ratio except the ratio of equality: and in each case there is only one such point of division. [Let AB be the given straight line and, since any given ratio THEOR. 3. A straight line which divides the sides of a triangle proportionally is parallel to the base of the triangle. [From Theor. 1 by the Rule of Identity, since by Theor. 2 there is only one point of division of a given line in a given ratio.] |