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494. If four magnitudes be proportionals, then, if the first be greater than the third, the second will be greater than the fourth; and if equal, equal ; and if less, less.
GIVEN, A:B ::C: D.
:. A:B > C:B; therefore, from our hypothesis,
C:D >C: B; therefore, by 491,
.. A:B :: C:B,
495. If four magnitudes of the same kind be proportionals they will also be proportionals when taken alternately.
Let A : B :: C : D, the four magnitudes being of the same kind, then alternately,
A : C:: B : D.
MA : mB :: n C : nD; therefore, by 494, mA >, =, or < nC, as mB>,=, or <nD; and, this being true for all values of m and n,
A : C:: B: D.
496. If two ratios are equal, the sum of the antecedent and consequent of the first has to the consequent the same ratio as the sum of the antecedent and consequent of the other has to its consequent.
If A:B :: C:D, then
A + B : B :: C + D : D.
PROOF. Whatever multiple of A + B we choose to examine, take the same multiple of A, say 17A, and let it lie between some two multiples of B, say 23B and 24B; then, by hypothesis, 17C lies between 23D and 24D.
Add 17B to all the first, and 17D to all the second ; then 17(A + B) lies between 40B and 41B, and 17(C + D) between 40D and 41D; and in the same manner for any other multiples.
1. Fundamental Geometric Proportions.
497. If two lines are cut by three parallel lines, the intercepts on the one are to one another in the same ratio as the corresponding intercepts on the other.
HYPOTHESIS. Let the three parallols AA', BB, CC', cut two other lines in A, B, C, and A', B', C', respectively.
CONCLUSION. AB : BC :: A'B' : B'C'.
mAB, and, in the same way, BN = n.BC, taking M and N on the same side of B. From M and N draw lines || AA', cutting A'B'C' in M' and N'.
B'M' = m. A'B', and B'N' n. B'C'. (227. If three or more parallels intercept equal sects on one transversal, they
intercept equal sects on every transversal.)
But whatever be the numbers m and n, as BM (or m . AB) is >, =, or < BN (or n.BC), so is B'M' (or m. A'B') respectively >, =, or < B’N (or n. B'C');
.:. AB : BC :: A'B' : B'C''.
498. REMARK. Observe that the reasoning holds good, whether B is between A and C, or beyond A, or beyond C.
499. COROLLARY I. If the points A and A coincide, the figure ACC' will be a triangle; therefore a line parallel to one side of a triangle divides the other two sides proportionally.
500. COROLLARY II. If two lines are cut by four parallel lines, the intercepts on the one are to one another in the same ratio as the corresponding intercepts on the other.
501. If a sect AB is produced, and the line cut at a point P outside the sect AB, the sect AB is said to be divided externally at P, and AP and BP are called External Segments of AB.
In distinction, if the point P is on the sect AB, it is said to be divided internally.
502. A given sect can be divided internally into two segments having the same ratio as any two given sects, and also externally unless the ratio be one of equality; and, in each case, there is only one such point of division.
GIVEN, the sect AB.
On a line from A making any angle with AB, take AC and CD equal to the two given sects. Join BD.
Draw CF || DB and meeting AB in F. By 497, AB is divided internally at F in the given ratio.
If it could be divided internally at G in the same ratio, BH being drawn || CG to meet AD in H,
AG would be to GB as AC to CH, and therefore not as AC to CD. (476. The scale of relation of two magnitudes will be changed if one is altered in
size:ever so little.)
Hence F is the only point which divides AB internally in the given ratio.