PROPOSITION 22. THEOREM. If there be any number of magnitudes, and as many others, which taken two and two in direct order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. = First, let there be three magnitudes A, B, C, and other three D, E, F, such that A: B it is required to prove A: C D: E, and B: C = E: F: = D: F. Of A and D take any equimultiples whatever mA, mD; of B and E any whatever nB, nE; and of C and F any .. according as mA is greater than qC, equal to it, or less, mD is greater than qF, equal to it, or less, V. 20 V. Def. 5 Second, let there be four magnitudes A, B, C, D, and other four E, F, G, H, such that A: B = E: F, Since A, B, C are three magnitudes, and E, F, G, other three, which, taken two and two in direct order, have the same ratio, Similarly the demonstration may be extended to any number of magnitudes. PROPOSITION 23. THEOREM. If there be any number of magnitudes, and as many others, which taken two and two in transverse order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. = First, let there be three magnitudes A, B, C, and other three D, E, F, such that A : B E: F, and B: C = D : E : it is required to prove A: CD: F. ន Of A, B, and D take any equimultiples mA, mB, mD; and of C, E, and F take any equimultiples nC, nE, nF. .. according as mA is greater than nC, equal to it, or less, mD is greater than nF, equal to it, or less; .. A: C = D: F. V. 21 V. Def. 5 Second, let there be four magnitudes A, B, C, D, and other four E, F, G, H, such that A: B = G: H, Since A, B, C are three magnitudes, and F, G, H other three, which, taken two and two in transverse order, have the same ratio, Similarly the demonstration may be extended to any number of magnitudes. COR. From this proposition and the preceding it may be inferred that ratios which are compounded of equal ratios are equal to one another. If the first has to the second the same ratio which the third has to the fourth, and the fifth to the second the same ratio which the sixth has to the fourth; the first and fifth together shall have to the second the same ratio which the third and sixth together have to the fourth. But again, E: B=F:D; A+E:BC+F: D, by direct equality. Hyp. V. 22 The terms of a proportion are proportional by addition and B: B = C Because A But A+ B: A B = equality. C + D : C - D, by direct [Proposition 25 has been omitted, as being of little use.] V. A V. 18 V. 22 286 BOOK VI. DEFINITIONS. 1. Similar rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportional. Of the two requisites for similarity among figures, namely, equiangularity and proportionality of sides, it will be seen from VI. 4, 5, that if two triangles possess the one, they also possess the other. In this respect triangles are unique. Hence, in order to prove two rectilineal figures (other than triangles) similar, it must be shown that they possess both requisites. 2. When any proportion is stated among the sides of two similar figures, those pairs of sides which form antecedents or consequents of the ratios are called homologous sides. 3. Similar figures are said to be similarly described upon given straight lines when the given straight lines are homologous sides of the figures. 4. When two similar figures have their homologous sides parallel and drawn in the same direction, they are said to be similarly situated; when they have them parallel and drawn in opposite directions, they are said to be oppositely situated. 5. Triangles and parallelograms which have their sides about two of their angles proportional in such a manner that a side of the first figure is to a side of the second, as the other side of the second is to the other side of the first, are said to have these sides reciprocally proportional. |