Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/188

164 PROPOSITION 19. THEOREM.

If a whole magnitude he to a whole as a magnitude taken from the first is to a magnitude taken from the other, the remainder shall he to the remainder as the whole is to the whole.

Let the whole AB be to the whole CD as AE, a magnitude taken from AB, is to CF, a magnitude taken from CD: the remainder EB shall be to the remainder FD as the whole AB is to the whole CD.

For, because AB is to CD as AE is to CF, [Hypothesis. therefore, alternately, AB is to AE as CD is to CF. [V. 16. And if magnitudes taken jointly be proportionals, they are also proportionals when taken separately; [V. 17. therefore EB is to AE as FD is to CF; therefore, alternately, EB is to FD as AE is to CF. [V. 16. But AE is to CF as AB is to CD; [Hyp. therefore ED isto FD as AB is to CD. [V.ll.

Wherefore, if a whole &c.

. If the whole be to the whole as a magnitude taken from the first is to a magnitude taken from the other, the remainder shall be to the remainder as the magnitude taken from the first is to the magnitude taken from the other. The demonstration is contained in the preceding.

PROPOSITION E. THEOREM.

If four magnitudes he proportionals, they shall also be proportionals by conversion; that is, the first shall be to its excess above the second as the third is to its excess above the fourth.

Let AB be to BE as CD is to DF: AB shall be to AE as CD is to CF.