Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/195

Rh PROPOSITION 24. THEOREM.

If the first have to the second the same ratio which the third has to the fourth, and the fifth have to the second the same ratio which the sixth has to the fourth, then the first and fifth together shall have to the second the same ratio which the third and sixth together have to the fourth.

Let AB the first have to C the second the same ratio which DE the third has to F the fourth; and let BG the fifth have to C the second the same ratio which EH the sixth has to F the fourth: AG, the first and fifth together, shall have to C the second the same ratio which DH, the third and sixth together, has to F the fourth.

For, because BG is to C as EH is to F, [Hypothesis. therefore, by inversion, C is to BG as F is to EH [V. B. And because AB is to C as DE is to F, [Hypothesis. and C is to BG as F is to EH; therefore, ex aequali, AB is to BG as BE is to EH. [V. 22. And, because these magnitudes are proportionals, they are also proportionals when taken jointly; [V. 18. therefore AG is to BG as DH is to EH. But BG is to C as EH is to F;[Hypotheseis. therefore, ex aequali, AG is to C as DH is to F. [V.22

Wherefore, if the first &c.

1. If the same hypothesis be made as in the proposition, the excess of the first and fifth shall be to the second as the excess of the third and sixth is to the fourth. The demonstration of this is the same as that of the proposition, if division be used instead of composition.

2. The proposition holds true of two ranks of magnitudes, whatever be their number, of which each of the first rank has to the second magnitude the same ratio that the corresponding one of the second rank has to the fourth magnitude; as is manifest.