Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/196

172 PROPOSITION 25. THEOREM.

If four magnitudes of the same kind be proportionals, the greatest and least of them together shall he greater than the other two together.

Let the four magnitudes AB, CD, E, F be proportionals; namely, let AB be to CD as E is to F; and let AB be the greatest of them, and consequently F the least: [V.A, V.14.

AB and F together shall be greater than CD and E together.

Take AG equal to E, and CH equal to F. Then, because AB is to CD as E is to F, [Hypothesis. and that AG is equal to E, and CH equal to F; [Construction. therefore AB is to CD be AG is to CH. [V. 7, V. 11. And because the whole AB is to the whole CD as AG is to CH; therefore the remainder GB is to the remainder HD as the whole AB is to the whole CD. [V. 19. But AB is greater than CD; [Hypothesis. therefore BG is greater than DH. [V. A. And because AG is equal to E and CH equal to F, [Constr. therefore AG and F together are equal to CH and E together. And if to the unequal magnitudes BG, DH, of which BG is the greater, there be added equal magnitudes, namely, AG and F to BC, and CH and E to DH, then AB and F together are greater than CD and E together.

Wherefore, if four magnitudes &c.