Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/163

Rh Let AB the first be the same multiple of C the second, that DE the third is of F the fourth, and let BG the fifth be the same multiple of C the second, that EH the sixth is of F the fourth: AG the first together with the fifth, shall be the same multiple of C the second, that DH, the third together with the sixth, is of F the fourth.

For, because AB is the same multiple of C that DE is of F, as many magnitudes as there are in AB equal to C,so many are there in BE equal to F. For the same reason, as many magnitudes as there are in BG equal to C, so many are there in EH equal to F.

Therefore as many magnitudes as there are in the whole AG equal to C, so many are there in the whole DH equal to F. Therefore AG the same multiple of C that DH is of F.

Wherefore, if the first de the same multiple &c.

. From this it is plain, that if any number of magnitudes AB, BG, GH be multiples of another C; and as many DE, EK, KL be the same multiples of F, each of each; then the whole of the first, namely, AH, is the same multiple of C, that the whole of the last, namely, DL, is of F.

PROPOSITION 3. THEOREM.

If the first he the same multiple of the second that the third is of the fourth, and if of the first and the third there he taken equimultiples, these shall he equimultiples, the one of the second, and the other of the fourth.