PROPOSITION 7. "That magnitude which has a greater ratio than another has to the same magnitude is the greater of the two; and that magnitude to which the same has a greater ratio than it has to another magnitude is the less of the two." That is, A, B, C being three magnitudes of the same kind; A : C > B: C, then A > B. if and if C: A > C: B, then A < B. Proof of Propositions 6 and 7. This proves the first part of each of the two propositions. Second Part: It has been proved that This proves the second part of each of the two propositions. (i). COROLLARY: If A: C> B: C, then C: A <C: B, and conversely. (ii). COROLLARY: If A: C>P: R, then C: A<R:P, and conversely. PROPOSITION 8. "Magnitudes have the same ratio one to another which their equimultiples have." Let A, B be two magnitudes of the same kind and m any integer; then AB: mA: m B. For if p, q be any two integers, m.pA> or <m.qB = (i). COROLLARY: If A: B :: P: Q, then mA: m B :: nPnQ, whatever integers m and n may be. PROPOSITION 9 "If two ratios be equal, and any equimultiples of the antecedents and also of the consequents be 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:n B:: mP:n Q, m and n being any integers. For, if, q be any two integers, then, since by hypothesis PROPOSITION 10. "If four magnitudes of the same kind be proportionals, the first is greater than, equal to, or less than, the third, according as the second is greater than, equal to, or less than, the fourth." then A> or < C according as B >= or < D. but If B > D, then A: BA: D; A B C D, (Prop. 5.) (Hypoth.) ... C:D<A : D, .'. A:D>C : D, (Prop. 7.) "If four magnitudes of the same kind be proportionals, the first will have to the third the same ratio as the second to the fourth." (Alternando.) Let A: B:: C: D, then will A: C :: B : D. (Hypoth.) (Prop. 7.) For m and n being integers, and A: B: mA: mB C:D::nC:nD, ... mA: mB:: nC: nD ... mA> or <nC according as mB> But m and n are any integers ; .. A: C: B : D. (Prop. 8.) (Prop. 1.) or <nD. (Prop. 10.) (Def. 5.) D.-SEVEN FUNDAMENTAL Theorems in PROPORTION.* PROPOSITION 12: (LEMMA).** "If on two straight lines, AB, CD, cut by two parallel straight lines AC, BD, equimultiples of the intercepts respectively be taken; then the line joining the points of division will be parallel to AC, or BD." On AB and CD, produced either way, let the respective equimultiples B E, DF of A B, CD be taken, on the same side of BD; then EF is parallel to BD. Since the For, join A D, DE, BC, B F. triangles ABD, CBD are on the same base BD, and their vertices A, C are in the line AC parallel to BD, they are equal in area; and whatever multiple BE is of AB, or DF of CD, the triangle DBE is that same multiple of the triangle ABD, and the triangle DBF of the triangle CBD. ... area of triangle EBD = area of triangle FBD. But these triangles EBD, FBD have the same base BD; hence their vertices E and F B F Fig. 3 must be in a straight line parallel to BD and therefore EF is parallel to BD. * Enunciations of Propositions 13-17 quoted from the Syllabus of Plane Geometry, Book IV, Section 2. ** J. M. Wilson: Elementary Geometry, page 205. PROPOSITION 13. "If two straight lines be 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 straight lines A A', BB', CC' be cut by two other straight lines A C, A'C' in the points A, B, C and A', B', C' respectively; then AB: BC: A' B': B' C'. = n. B'C', Also on A For, on AC take B M = m. AB, BN n. BC, m and n being integers, M and N on the same side of B. A'C' take B' M' =m. A'B', B'N': M' and N' being on the same side of B' as M and N of B. Then, by the foregoing lemma (Prop. 12), M M' and N N' are both parallel to B B' and cannot meet. Hence, whatever integers m and ʼn may represent, B' M' (or m. A'B')>= or < B'N' (or n. B'C') according as B M (or m. A B) >= or < BN (or n. BC); ... AB: BC:: A'B': B'C'. B N Fig. 4 (i). COROLLARY: "If the sides of a triangle be 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." (ii). COROLLARY: "If two straight lines be cut by four or more parallel straight lines, the intercepts on the one are to one another in the same ratio as the corresponding intercepts on the other." (iii). COROLLARY: If in any triangle, as OAB, a straight line EF, parallel to the side A B, cut the other sides, OA in E and OB in F, then AB: EF: OA:OE:: OB OF. |
